I created this guide as all guides I found have no actual explanation of the achievements and just repeat what the achievement says. Sometimes, that is enough. Sometimes, it's not.
If anything is wrong, or you think should be mentioned, do let me know and I will add it.
Achievements are listed in rarity order, as there are so many of them.

There are 3 DLCs with their own achievements: Puppies & Kitties, Horse Shelter, and Vet Clinic.

Some of the achievements seem to overlap. As I got all 3 DLC at once, I am not sure what may need multiple DLC and what can be achieved without. If I need to make a note on any of them, do let me know.

There are several broken achievements. Great, I know. Some of them are locked behind uncertain/inconsistent unlocks and some are just completely broken.
Looking for cosmetlogist/beauty salon? Look in the horses section. It will explain why.

There should be no more achievements that are completely unobtainable. At the time of this update, the riding/walking areas are not accessible [they give a game over because the area sets your rank and money to 0]. I do not know if that has broken any of the dog walking achievements completely, as riding horses in the paddock works and therefore walking dogs in the run should theoretically also work. Does it? I do not know, I cannot test it as I have these achievements already.

This game has DLC, each of which have their own achievements. These should all be achievable in the base game, without any DLC content. That means it will be in the main shelter, not the horse shelter.


It'll be allright!
Your first pet on board
This is the most common achievement. Follow the tutorial and you will achieve it regardless of start. It is given when you bring the first pet into your shelter.

First paycheck
Your first $1000
This is given to you when you have $1000 in your bank account.

Buddies
Two pets - two times the mess
This is given to you when you have at least two pets in your shelter at the same time.

What a clever student!
Now you know exactly what to do
This is given to you during the tutorial, after adopting out the pets.

New career
Your first star
As you adopt out pets, you get reputation. This achievement is given after you have enough reputation for a 1 star shelter.

Additional lawn
First upgrade, a little more space to run
This is given when expanding the shelter. Expansion tickets are given with reputation and stars. Using the first ticket will give you this achievement.

New employee
First hour of work!
Play the game an hour. Woo!

Like a donut in butter
There's nothing better than family
Adopt out a dog. If you start with cats, it'll take a moment but it won't be too long.

Responsible caretaker
You're giving a good example with cleaning that poop
Pick up one of the pet's poop. The bag dispensers, which come with furnished runs and some cages, give free poop bags for this. You can technically buy them if need be.

Our small house
5th pet with a chance of a better life
Adopt out 5 pets.

A home without a cat is just a house
Finally a new home
Adopt out a cat. If you start with dogs, it'll take a moment but it won't be too long.

Just five more minutes, mom!
Excuse me, who gave you permission to wake this baby up?!
When a pet is sleeping, pick it up or pet it. It'll force it to take up.

Beginner photographer
5 first photos - your career starts now!
Take 5 photographs with any camera.

Washy-washy!
First wash with no complications
Dogs are the only pet in the main shelter that are washed. Buy a wash room and bring in the dog to the pad pool. It comes equipped with the tools otherwise you can buy them. The steps, shown in game, are shampoo, sponge, wash off with water, and then dry.

Safe spot
10 happy families
Adopt out 10 pets.

Bad people
You did what you considered the best for the animal. It'll be safer under your care
When adopting out pets, some people are not good fits and can be discarded. This achievement is given the first time you press the red X and discard an adoption submission.

Builder
3 buildings at 3rd level!
Each building you can build has a different level. Most levels are locked behind reputation and a higher price tag. After placing three buildings at max level [3], you will get this achievement.

Well-bred
This animal is almost perfect! How did you manage to achieve it?!
This is done by perfectly matching up desired traits, undesired traits, and best conditions for the pet. It is based on luck how quickly you get this. If someone does not match the pet, you can discard them and wait on another submission. After too long, the ad will have to be reposted.

It's just a beauty mark
...but why is it moving?
When a sick animal is taken in, there is a chance they have a tick. Remove it for this achievement. This requires a vet clinic and tweezers.

Catch!
Good boi! 15 catches!
If you do not have a ball, you will need to buy it. Play fetch 15 times by throwing the ball with one of your dogs and you will get this achievement.

Pat pat!
How nice, 10 pat-pats!
Petting is the easiest way to increase fun. Pet an animal 10 times for this achievement.

Regular customer
You've ordered 10 packages!
Order 10 separate times from the buy menu, and have the package delivered, and you will unlock this achievement.

Animal Doctor
You've done all the types of treatment. Now you're a real expert!
I believe this can be achieved with the base game clinic. The treatments include: removing a tick, applying a bandage, and shaving a wound to apply and later remove plaster.

Animal lover
After five hours you look very professional
Play the game for 5 hours.

Start small
50 animals were adopted from all your shelters!
Adopt out a total of 50 animals. Any kind will do.
Note: I am decently sure this achievement is bugged and will be given out earlier than 50 animals.

Sandpit
No one likes the toilet smell, so you cleaned your first litterbox
Only cats use the litter box. Place one in a cat's cage and fill it with litter. After they use it 3 times, it will have a stink cloud. Grab a poop bag and remove all of the poop. Remember to fill it up again with litter afterward :)

Shopaholic
You really bought an item of each type from the shop! What will you do with them?
Buy an item for each of the tabs [Food, Hygiene, Toys, etc] for this achievement.

Shelter resident
10 hours?! Are you living here or what?
Play the game for 10 hours.

It's getting louder
3 stars! Even the newspapers are mentioning us!
As you adopt out, your reputation will increase. Get enough reputation for a 3 star shelter for this achievement.

First walk
The first walk on a leash is always the most memorable!
Only dogs can go for a walk. Buy a leash and use it on a dog. The dog will now follow you. Take it to the glowing circle at the front gate. It will ask you to go to the dog park. Click yes to take your dog on a walk in the dog park. You can return via another glowing circle at the front of the dog park.

First-class patting
20 animals pet. It's so nice, feels so much better
Pet 20 animals.

Safe space
30th pet under your care. You're a really good person!
Take in 30 animals. Any kind will do.

Express wash
You washed a dog under 1 minute? Let's hope it was done properly?
Complete the entire wash cycle on a dog in 1 minute. It's really lucked based on if a wash will register quickly or not. Sometimes they don't want to clean and sometimes they do quickly.
Note: I have been informed you do not have to actually finish the full wash [i.e. fill the circle] for this achievement. Just start the process and move on (Thank you Kandle for the information).

Are you proud of yourself?!
What kind of caretaker loses their pet...
Don't take care of the pet. Eventually, it will be taken away. This is most common when an animal gets sick and you don't treat/don't have the things to treat it.

Great friends
5 snacks!
You can buy a basic or super snack. Feed them to your animals as you see fit. Do this 5 times for the achievement.

I'm just getting started
100 animals were adopted from all your shelters!
Adopt out a total of 100 animals. Any kind will do.
Note: I am decently sure this achievement is bugged and will be given out earlier than 100 animals.

Waste export
Where's the garbage truck? We've got 25 carton boxes to utilize!
Buy things from the shop 25 times and toss the boxes in the rubbish bin after you empty it out.

Developer
You've built all of the building types!
Build from each tab of buildings: Cages, Office, etc.

Sand cake
Woah, 10 litterboxes! That's plenty of cleaning, but satisfaction is guaranteed!
Clean a dirty litter box 10 times. Remember to fill it up with litter after you clean it.

The happy ending
50 animals with new lives - all thanks to you!
Adopt out 50 cats and/or dogs. "Start Small" seems to be any animals, and this one just base animals.

Villa with a pool
The biggest upgrade! What are you going to do with such a big lot?!
As you expand to fit in more buildings, you will get this achievement. It stats that the building was for a max lot, but I got it well before I ran out of expansion tickets.

Bankrupt
Save up, they said... Now you have nothing!
Have $0. I got this achievement by bringing a dog to the dog park on a leash as you will be set to having nothing. If that doesn't work, just buy until you have no money. Probably want to do this on a new save or load it after.

A true philantropist!
150 animals were adopted from all your shelters! That's a lot of happy new homes!
Adopt out 150 animals. I do not remember if this one is also broken, but adopting out any 150 animals should get you the achievement.

Pro photographer
50 photos?! You've got some experience, indeed!
Take 50 photos with any camera. As you take photos to adopt out animals, it should give you this achievement. Use the digital camera to automatically upload the photos for an easier time.

Three's not a crowd
You have 3 cats and 3 dogs in your shelter!
Have 3 cats and 3 dogs simultaneously. I believe they have to be adults, not puppies or kittens, if you do have that DLC.

At full capacity
10 pets at once makes the shelter really crowdy. I'm shocked you're still keeping it together! Good job
Have 10 animals of any kind in the shelter at the same time.

Regional 1st Place
You achieved 1st place in your region
As you gain reputation, your region rank will increase. These are not real people, just goals to meet. Get enough reputation to be 1st in your region and you should unlock this achievement. You should also get a trophy via the drone automatically delivered to you.

Sand castle
20 litterboxes clean. Great collection of treasures!
Clean 20 litter boxes. Don't forget to refill them once they are clean.

Free shipping
You've ordered 50 packages!
Order 50 separate purchases from the shop. You can spam one snack 50 times if you really want, but I got this easily enough just by playing through the game.

A short walk
Go for a walk with an animal for 5 minutes.
Take a dog on a leash to the dog park for a total of 5 minutes.

Real animal lover
You could use a third hand. You pat-patted 50 animals!
Pet 50 animals.
Note: I do not know if you can spam this on one animal 50 times. As you work through the game, you will see more than 50 animals, so you can do it separately easily enough.

Serious business
Wow, it's $50.000 already!
Have $50,000 in your bank account simultaneously. This is not cumulative over time, you have to have it all at the same time.

A little bit of both
You have 5 cats and 5 dogs in your shelter!
Have 5 cats and 5 dogs simultaneously. I believe they have to be adults, not puppies or kittens, if you do have that DLC.

Spring cleaning
Cleaning control over 95%
When you get an email of an incoming inspection, you will need to have your cleanliness in your shelter over 95%. You can find you current cleanliness level under stats. The easiest method is to have no pets in the shelter until the inspection. Otherwise, I set all my employees to clean and place the animals in the run [on the grass] so they do not dirty the floors.

Photomodel
After 10 photos this animal can be a real top model
Have 10 photos of the same animal. You can spam it if you want. If you pose them if you don't. Up to you. Any animal type will work.

Everybody knows me
Maximum reputation! Animals dream of being under your care!
Have a 5 star shelter. This is achieved by having a certain amount of reputation.

Dot hunting
Eat the dot, eat the dot... at least 10 times!
This is done using a laser pointer and a cat. If you keep the pointer still, the cat will do a weird broken animation of trying to bite the laser. This needs to be done 10 times for the achievement. While it can be done with just cats, I found it easiest with kittens.

Getting crowded
100 animals have walked through your shelter! AMAZING!
Take in 100 different animals. Any kind will do.

Rich guy
$100.000?! How are you going to spend that?
Have $100,000 in your bank account simultaneously. This is not cumulative over time, you have to have it all at the same time.

Know-it-all
Now you know everything, the game is an open book for you
THIS CANNOT BE DONE WITH THE CAT START. The expansion encyclopedia portion is bugged out and will not unlock if you start with cats. It will only work for dogs. There are encyclopedia portions for each DLC, but I do not know if they will show if you do not have the DLCs. If they do, you will need all the DLC to get them.
Note: The "Treating puppies and kittens" portion is unlocked when you buy and pick up cough syrup for puppies and kittens. Why that specifically? No idea.

Snack guy
30 snacks? You know they're gonna be fat, right?
You can buy a basic or super snack. Feed them to your animals as you see fit. Do this 30 times for the achievement.

Foodie
You've emptied 10 food packs!
Completely empty out any 10 food packs. Does not matter what level or for what animal. If you want to spam it, fill a bowl with one kind of food then immediately switch to another kind of food and it will empty out.

Impulse buyer
You've ordered 100 packages! Your wallet seems slimmer.
Order 100 separate purchases from the shop. You can spam one snack 100 times if you really want, but I got this easily enough just by playing through the game.

A stroll
Travel 1000 meters with an animal on a leash.
While walking a dog on a leash in the dog park, walk for a total of 1000 meters. This does not have to be all at once, as it is cumulative.

National 1st Place
You achieved 1st place in your country
NOTE: I have heard from multiple people that this achievement is bugged. Save your game before you get the rank. After the trophy is delivered and grabbed, if you did not get the achievement, try to send out a few more animals . If that does not work, I suggest loading your save and trying again.
As you gain reputation, your country rank will increase. These are not real people, just goals to meet. Get enough reputation to be 1st in your country and you should unlock this achievement. You should also get a trophy via the drone automatically delivered to you.

Medium walk
Go for a walk with an animal for 30 minutes.
While walking a dog on a leash in the dog park, walk for a total of 30 minutes. This does not have to be all at once, as it is cumulative.

You've made their day!
Make 10 animal happy by walking them.
This achievement is fairly buggy. The most consistent way I have found to have it work is to take a dog with their fun in red [negative] and take them for a walk until it is 3/4s of the way filled [positive and green]. This does not always count, but usually does. I also detach and reattach the leash a few times before leaving the dog park as it often bugs out when I get back to the shelter. This needs to be done 10 times at least.

A hike
Travel 3000 meters with an animal on a leash.
While walking a dog on a leash in the dog park, walk for a total of 3000 meters. This does not have to be all at once, as it is cumulative.

Decorator
50 decorations in your shelter! Well done, it's so pretty now!
From my attempts to do this with furniture, it seems you may have to pick from the decorations tab while building. Place 50 of any of these for the achievement. I used lamp posts to use up the least amount of space.

The clean guy
You scored 95% for the 5th time! It's so clean you can eat straight off the floor
When you get an email of an incoming inspection, you will need to have your cleanliness in your shelter over 95%. You can find you current cleanliness level under stats. The easiest method is to have no pets in the shelter until the inspection. Otherwise, I set all my employees to clean and place the animals in the run [on the grass] so they do not dirty the floors.
I got 2 scores per inspection. Not sure if that was for each type of cages [dogs and cats] or if the puppies & kittens building was separate. Whatever it was, it makes getting all 5 a little easier.

A proper walk
Go for a walk with an animal for 60 minutes.
While walking a dog on a leash in the dog park, walk for a total of 60 minutes. This does not have to be all at once, as it is cumulative.

Hefty eater
You've emptied 30 food packs!
Completely empty out any 30 food packs. Does not matter what level or for what animal. If you want to spam it, fill a bowl with one kind of food then immediately switch to another kind of food and the bowl will empty out.

Perfect balance
You have 10 cats and 10 dogs in your shelter!
Have 10 cats and 10 dogs simultaneously. I believe they have to be adults, not puppies or kittens, if you do have that DLC.

A trip
Travel 5000 meters with an animal on a leash.
While walking a dog on a leash in the dog park, walk for a total of 5000 meters. This does not have to be all at once, as it is cumulative.

Glutton
You've emptied 50 packs of food! Who's eaten all that?
Completely empty out any 50 food packs. Does not matter what level or for what animal. If you want to spam it, fill a bowl with one kind of food then immediately switch to another kind of food and the bowl will empty out.

World 1st Place
You achieved 1st place in the world!
NOTE: I have heard from multiple people that this achievement is bugged. Save your game before you get the rank. After the trophy is delivered and grabbed, if you did not get the achievement, try to send out a few more animals . If that does not work, I suggest loading your save and trying again.
Second Note: I got a "rank up, #1 global" notification well before I hit the 100k points needed. I did not get the achievement at that time, but I did when I got the proper 100k points + adopted out one more animal.
As you gain reputation, your world rank will increase. These are not real people, just goals to meet. Get enough reputation to be 1st in the world and you should unlock this achievement. You should also get a trophy via the drone automatically delivered to you.

Luxury Stud
All buildings are at the highest level!
For every tab of building you own (Cages, Office, etc), the buildings on the map must be at the highest level available in that tab. It's level 3 for most tabs, but sometimes it only goes to level 2.
This achievement used to be broken. I got it as soon as I signed in after the fix, so it appears to work. That being said, I could not experiment to see which tabs are relevant. I know I only had the DLC vet building and not the base game vet building so theoretically you do not need both.

Something to note: for the hoof filing achievements, this is a rare ailment. I would advise saving it when one comes in and then treating the horse. Instead of saving again, load it so they are sick with overgrown hooves again. Do this until you are either tired of it or have all hoof related achievements. Doing this will also give you progress in the cure ailments achievements.

First visit at the stable
Enter the stable for the first time.
When facing the front gate of the main shelter, you can click it to enter the horse stable. The first time you do that, you should get the achievement.

First Horse
First horse at the shelter!
Bring your first horse into the shelter.

A free spirit
Find a home for 1 horse.
Adopt out your first horse.

Good as new
Heal 1 horse ailment.
Heal one horse of any ailment.

Wild like thunder
Find a home for 10 horses.
Adopt out 10 horses. Do they unsettle you like they do me?

Racing with eagles
Find a home for 50 horses.
Adopt out 50 horses.

All better now
Heal 15 horse ailments.
Heal 15 horses of any ailment. This is easiest done by just adopting them when they are sick.

First hoof filing
File hooves once
Using a hoof file, which you have to buy, fix the ailment of overgrown hoof once.

Please don't kick, please don't kick...
File hooves 10 times
Using a hoof file, which you have to buy, fix the ailment of overgrown hoof ten times. Again, I advise saving and loading for the same horse as it is not a common ailment.

Cosmetlogist Yes, it is misspelled in game.
Make 10 perfect hits while clipping claws or filing hooves
Making 10 perfect hits does NOT unlock this achievement. I even did 10 in a row. Nothing. I DID get this unlocked at the exact same time as I got "File hooves 10 times". I would put that under coincidence, if it did not also happen with "file hooves 30 times."

Healthy as a horse
Heal 50 horse ailments.
Heal 50 horses of any ailment. This is easiest done by just adopting them when they are sick.

Trot
Ride a horse for 1 hour.
You need both a saddle and a bridle on the horse to ride. You have to take off the harness to put the bridle on. This is achievement is unlocked after riding horses for a total of 1 hours. This is cumulative and does not have to be all at once.
Riding in the horse paddock DOES count for the achievement. I find this easier than going to the "proper" riding area as you can stop to feed/water/clean your horse as needed.

Canter
Ride a horse for 2 hours.
This is achievement is unlocked after riding horses for a total of 2 hours. This is cumulative and does not have to be all at once.
Riding in the horse paddock DOES count for the achievement. I find this easier than going to the "proper" riding area as you can stop to feed/water/clean your horse as needed.

These things really do grow quickly, huh?
File hooves 30 times
Using a hoof file, which you have to buy, fix the ailment of overgrown hoof 30 times. Again, I advise saving and loading for the same horse as it is not a common ailment.

Beauty salon
Make 30 perfect hits while clipping claws or filing hooves
Making 30 perfect hits does NOT unlock this achievement. I made far more than 30 perfect hits with claws. Nothing. I DID get this unlocked at the exact same time as I got "File hooves 30 times". I would put that under coincidence, if it did not also happen with "file hooves 10 times."

Gallop
Ride a horse for 5 hours.
This is achievement is unlocked after riding horses for a total of 5 hours. This is cumulative and does not have to be all at once.
Riding in the horse paddock DOES count for the achievement. I find this easier than going to the "proper" riding area as you can stop to feed/water/clean your horse as needed.

First puppy adopted
A puppy has been adopted for the first time!
Adopt out your first puppy. Fairly straight forward.

First kitten adopted
A kitten has been adopted for the first time!
Adopt out your first kitten. Fairly straight forward.

All patched up!
You've healed 5 kittens or puppies
Heal 5 kittens or puppies of any ailment.

Bubble bath
Cleaned 5 puppies or kittens
Clean 5 kittens or puppies. This is done with a cleaning cloth, which you have to buy.

Healing touch
You've healed 10 kittens or puppies
Heal 10 kittens or puppies of any ailment.

A true veterinarian
You've healed 15 kittens or puppies
Heal 15 kittens or puppies of any ailment.

Clean as a hound's tooth
Cleaned 10 puppies or kittens
Clean 10 kittens or puppies. This is done with a cleaning cloth, which you have to buy.

Living in luxury
You've bought the best possible building for your puppies and kittens!
Buy the level 3 building for the puppies and kittens. It combines the cages for the puppies and kittens into one building.

A professional groomer
Cleaned 15 puppies or kittens
Clean 15 kittens or puppies. This is done with a cleaning cloth, which you have to buy.

Full house
Your shelter is filled to the brim with baby animals!
I am honestly not sure exactly what is required for this. Apparently, other people are also have having issue with it. I think I had the level 1 building for puppies and level 2 for kittens and filled both with their respective animals (thank you for jogging my memory Your Schizophrenia). Mentioned user states they had 8 puppies and 5 kittens for it to unlock.
If you know exact parameters, feel free to let me know and I can update it.

Something to note: While the overgrown claws are not as rare as the horse overgrown hooves, they are not common. Having a save file to farm overgrown claw clipping may be a good idea.

First successful patient
Cure your first animal
Considering the percentage of completion, I believe this only unlocks by curing an ailment from the DLC. Cure any one animal with one of these ailments for the achievement.

Successful analysis
Analyze a sample
Using the sample analyzer, analyze your first sample.

Scanning...
1 ultrasound performed
Using the ultrasound, perform a scan on any animal. It has to be requested of you in order to perform.

First X-ray
Do an X-ray
Scan an animal for the first time with the x-ray. The X-ray machine has to be turned on before it can be used. Then it can be controlled with the computer on the wall outside of the room.

4x lens
Use the microscope once
Use the microscope for the first time. Samples must be placed on a cover slip to work. The low level microscope is awful. The better clinic comes with a better one, but I would not dismiss buying a better one as soon as possible.

First listen
Use the stethoscope once
Use the stethoscope on any animal once. I believe it has to be requested in order to perform.

A healthy troop
Cure 10 animals
Cure 10 animals of any of the DLC vet ailments. This can be done on any animal.

First clip
Clip claws once
Clip overgrown claws for the first time. This can be done on dogs, cats, puppies, or kittens.

10x lens
Use the microscope 10 times
Use the microscope ten times. Samples must be placed on a cover slip to work. The low level microscope is awful. The better clinic comes with a better one, but I would not dismiss buying a better one as soon as possible.

Manicurist
Make 5 perfect hits while clipping claws or filing hooves
For some reason, this one is not bugged. Perform 5 perfect hits for clipping or filing on any animal for the achievement. It is easier with the higher level clippers/file.

Just open up a hospital already
Cure 30 animals
Cure 30 animals of any of the DLC vet ailments. This can be done on any animal.

Basically a photographer
10 X-rays done
Scan an animal 10 times with the x-ray. The X-ray machine has to be turned on before it can be used. Then it can be controlled with the computer on the wall outside of the room.

Analysis master
10 samples analyzed
Using the sample analyzer, analyze 10 samples. I have found horses want disproportionately more analyzed samples. If you have the horse DLC, it is easier to farm with sick horses.

Hear that bass
Use the stethoscope 10 times
Use the stethoscope on any animal 10 times. I believe it has to be requested in order to perform.

Scanning some more...
10 ultrasounds performed
Using the ultrasound, perform a scan on 10 animals. It has to be requested of you in order to perform.

100x lens
Use the microscope 30 times
Use the microscope 30 times. Samples must be placed on a cover slip to work. The low level microscope is awful. The better clinic comes with a better one, but I would not dismiss buying a better one as soon as possible.

No more tippy-taps on the floor
Clip claws 10 times
Clip overgrown claws 10 times. This can be done on dogs, cats, puppies, or kittens. Again, I recommend save and loading on one animal as it is not a common ailment.

It's a necklace at this point
Use the stethoscope 30 times
Use the stethoscope on any animal 30 times. I believe it has to be requested in order to perform.

We're all spooky during an X-ray
30 X-rays done
Scan an animal 30 times with the x-ray. The X-ray machine has to be turned on before it can be used. Then it can be controlled with the computer on the wall outside of the room.

Will you ever stop scanning...
30 ultrasounds performed
Using the ultrasound, perform a scan on 30 animals. It has to be requested of you in order to perform.

A scientist
30 samples analyzed
Using the sample analyzer, analyze 30 samples. I have found horses want disproportionately more analyzed samples. If you have the horse DLC, it is easier to farm with sick horses.

Get rid of those weapons!
Clip claws 30 times
Clip overgrown claws 30 times. This can be done on dogs, cats, puppies, or kittens. Again, I recommend save and loading on one animal as it is not a common ailment.