Here are some explanations on terminology I will use in this guide:

Friendship points - points you gain with NPCs when interacting with them, required to level up friendships and increase hearts/stars. Also known as favor.

Knowledge points - points you gain when you level up any Knowledge skill (combat, gathering, workshop or social)

Perks - skills/abilities you unlock using Knowledge points.

Knowledge EXP - experience required to level up knowledge skills. Is separate from friendship points, character exp or workshop rank exp.

I would also like to clarify that you do not need to max out all Knowledge levels. Knowledge like Gathering only has 33 levels worth of perks while Combat only has 31. You only need to learn all perks in order to unlock the achievement despite both of those Knowledge skills continuing leveling up until level 37.

UPDATE: In regard to pets and EXP - according to other players pets hitting enemies or gathering things does not provide any experience so if you're trying to power-level avoid using them.

Knowledge EXP is gained through interactions with townsfolk or animals.

This is the hardest, most tedious Knowledge to level up because it has 37 levels worth of perks and you have to go completely out of your way to interact with people daily for a long time. It also doesn't make much sense since friendship points do not in any way affect Knowledge EXP gained and like half of the interactions don't even give any Social Knowledge EXP.
It is highly recommended to acquire Social Butterfly perk as soon as possible because it let's you sometimes gain double EXP from social interactions. Effects are random and don't seem to trigger very often but every little bit helps.

Here is a list of interactions that give you Social Knowledge EXP:

Talking - talking to NPCs and animals gives you 2 Social Knowledge EXP per conversation. Every NPC gives Knowledge EXP for talking to them once per day, even if there are events that let you gain more than one instance of friendship point increase these repeated dialogues WILL NOT give you any Social Knowledge EXP.

Gifts - giving gifts to NPCs and animals gains you 5 Social Knowledge EXP per gift. It does not matter if the gift is loved, liked or even hated by the NPC - you will always gain 5 Social Knowledge EXP. Additionally, even if the NPC does not accept the gift (for example, Fang never accepts party invitations) you will still gain 5 Social EXP for the attempt.

Group photos - Every photo you take with an NPC gives you 1 Social Knowledge EXP.

Sparring - sparring gives you 3 Social Knowledge EXP per fight, whether you win or lose. The plus side is you also gain Combat Knowledge EXP but I do not recommend sparring unless Social Knowledge leveling is all you do because every match takes 30 minutes out of your day.

Commissions - commissions are a great way to gain Social Knowledge EXP since they give you 10 Social EXP per commission completed. Both building commissions and requests (board requests for food, plants or fish) give you 10 Social EXP each so make sure to do them as much as you can.
Civil Corps commissions for killing monsters DO NOT give any Social Knowledge EXP.

Additionally, once you reach a certain point in the story you will be able to attend Burgess first sermon. Doing so will grant you 1 Social Knowledge point (perk point, not EXP point) thus allowing you to unlock all skills at Social level 36 instead of 37. The game says you get 5 Knowledge points but that seems to be a mistake since you only actually get one.

And that's it.
Dates/hangouts DO NOT give any Social EXP.
Critters minigame DOES NOT give any Social EXP.
Parties DO NOT give any Social EXP (giving invitations works the same as gifts, though).
Interactions like Observe, Hug, Pet animals and so on DO NOT give any Social EXP.
NPC sidequests DO NOT give any Social EXP.

Knowledge is gained by performing successful attacks.

There is 31 perk to be gained from Combat Knowledge tree and it is the second hardest Knowledge to level up. Combat EXP is gained per successful attack and not per enemy hit, meaning if you hit 1 enemy with 1 attack you will gain the same amount of EXP as if you hit several enemies with one attack. So you should avoid using AOE attacks and always try to only hit 1 enemy at a time. With this system you are basically being punished for getting stronger because killing enemies faster means getting less EXP from them.
Additionally, it seems like weapons that hit enemies multiple times per 1 attack give more Combat EXP than single hit weapons so bad news for Greatsword enjoyers like myself.

Here are some tips and tricks on how to efficiently gain Combat Knowledge EXP:

1. Breakable wooden boxes and barrels also give Combat EXP, not just the enemies. So smash them as you see them!

2. Try using weak weapons that deal multiple hits with a single attack like daggers. Avoid Greatswords, since they deal big damage and hit enemies once per attack (except last attack in the combo) they are terrible for Combat EXP farming (even though I like using them in general).

3. Make sure you don't have any attack boosting furniture in your house/property. That way you'll be killing enemies slower and gaining more EXP per enemy.

4. You might think that using bare fists would be the best since they deal the least amount of damage. That is wrong for two reasons:
a)Game has an annoying mechanic where consumables (like meat or honey) get auto-equipped into your quickslots if there is an empty one and fist attacks require an empty slot so it makes the whole thing very annoying.
b)Weaker enemies seem to only give 0.5 EXP per punch so you need to punch them twice to get 1 Combat EXP while stronger enemies give 1 EXP per punch. This inconsistency makes fist EXP farming worse, other single-hit weapons like Greatswords or Hands Of Light weapon do not behave like this (they always give 1 EXP per hit).

5. Only weapons work for Combat EXP. Attacking enemies with tools like an Axe or Pickhammer DOES NOT give you any Combat Knowledge EXP.

6. My personal favorite EXP farming weapon is Altair RX-67, a late game sabre-katana. It hits several times with every attack which gives 2 Combat EXP per successful hit (4 EXP on some specific enemies like Alpha Yakmels) and it also sends out energy waves so you can hit enemies from a distance. Good thing about those waves is that if they go through several enemies you will get EXP for every enemy connected unlike with melee attacks. Just be careful and only use the basic 2 attack combo since the third attack is a single-hit that deals massive damage.

7. If you have access to a late game dungeon Lab 7 (Dead Sea Ruins) it is a great place to farm Combat EXP. At Combat level 20+ even the first level of the dungeon gave me 2-3 levelups per run (level 54 character using Altair RX-67).

8. Saving the game and then re-loading seems to sometimes respawn enemies. Doesn't seem to always work but works better if you save-quit the game-load.

In case anyone was wondering, the Game Center punching minigame and Whack-a-mole DO NOT give Combat Knowledge EXP. I was curious so I tested it lol.
You also gain Combat Knowledge EXP from sparring with people but I don't think that is a good way to farm EXP so I would not suggest that.

Knowledge leveled up by producing things at your workshop.

This one is very straight forward - just make things at your worktable, machinery or assembly station. Every time you make one item at the worktable you gain 1 Workshop Knowledge EXP. Every item produced by machines also gives 1 Workshop EXP. Assembly Station gives you 9 Workshop EXP per item completed. Dew Collectors DO NOT give any Workshop Knowledge EXP. Growing produce on your property DOES NOT grant Workshop or Gathering Knowledge EXP. Refining items at the Refiner DOES NOT give any Workshop Knowledge EXP. Ore Refiner also DOES NOT give any Knowledge EXP.

This Knowledge tree should be no problem and it should level up naturally as you play, but in case you need extra info here it is:

1. Bulk crafting works as you would expect - 1 unit of water (for example) crafted = 1 Workshop Knowledge EXP point gained while 40 units of water crafted = 40 Workshop Knowledge EXP points gained.

2. Machinery gives you Knowledge EXP as you collect produced items. Assembly Station gives EXP when you collect finished item.

3. You might want to hold off from building a Factory because things produced from it don't seem to be giving Workshop Knowledge EXP, for some reason.

Knowledge levels up by gathering resources.

Just like with Workshop Knowledge, this one should level up naturally as you play. Simply gather plants or worms, kick trees, mine rocks, cut down trees (or don't, Burgess will be sad if you do) and burrow through abandoned ruins. SImilar to Combat EXP you gain Gathering Knowledge EXP for every "tick" meaning rocks that take more hits to break will grant you more Gathering EXP. Same with tools like Chainsaw or Drill, every "tick" gives you Gathering EXP.

In my tests hitting something with your Pickhammer, for example, granted 2 Gathering Knowledge EXP but I haven't tested absolutely everything so that might or might not be always consistent.
Collecting plants like Oregano gives you 2 gathering Knowledge EXP per tick, since you gather twice from one plant you get total of 4 Gathering EXP. (NOTE: I've been informed that one gathering tick for herbs only gives 1 EXP. Whether my 2 EXP were a result of perks triggering or something has been changed in patches I don't know, so experience may vary)
Collecting worms and manure gives 1 Gathering Knowledge EXP.

So, that is it. I hope you found this guide informative and helpful.

Happy building and remember - conserve water!