1. You will have to be in year 4 or past that. Dragon Eggs can only be found on the Kozita island, which is only available on year 4. And you will have to complete a questline to get to Kozita.

2. You will have to clear the thorns on Kozita. Try to complete the questline with Captain Conrad, and then help the Druid to clear the thorns.

3. You will need to complete a third questline, this time with the miners on Rock Island. After that there will be a quest called "Meet Mina" and you will have to talk to Mina to complete it. Then you will have to complete another quest from Mina involving a fake Dragon Egg.

There are Dragon Nests scattered all over Kozita. Sometimes, there will be a Dragon Egg in a nest. The Dragon Egg inside will be a Spring, Summer, Fall, or Winter egg depending on the season. There is a small chance you will find a Dark Dragon Egg or a Fairy Dragon Egg in a nest. This can happen in any season. After you find any Dragon Egg, go talk to Mina and she will give you a Hatchery. Place the hatchery in the world and put the Dragon Egg inside. After cuddling the egg every day for 14 days, it will hatch.

Mina will sometimes trade a Dragon Egg for 10 Dragon Statues. Just trade in the statues for the egg if this happens. Dragon statues can be found scattered around Kozita.

-Dragon Rock Statues (the statues outside of some caves that look like dragon's heads) will usually have their eyes "closed", but when a Dragon Egg spawns their eyes will "open up". They appear as white crystals.
-Dragon Eggs and the eyes of the statues spawn at 6:20, 20 seconds after the player wakes up.
-You can use camera mode to fly around and look for eggs without time passing. To enter camera mode, simply press F11 (F10 also enters camera mode but won't pause the game). To exit press F10. Camera mode is helpful because you can phase through objects and go almost anywhere.
-Mina will sell any Dragon Egg even if it's out of season. Mina's sold item is different for each player character.
-The Dragon Egg image at the front of the guide is a Summer Dragon Egg, so it's green. Spring Dragon Eggs are pink, the fall/autumn ones are orange, and winter eggs are light blue. Dark eggs are black and the fairy ones are pastel-colored.

Now for some Dragon stuff...

Dragons have eight types. The seasonal types, Spring, Summer, Fall and Winter appear based on the season, but Dark and Fairy Dragons appear in any season (they are a bit rarer). Well, actually their eggs are what appear. The Dragons themselves don't appear in-game unless you hatch a Dragon Egg... do Dragons even lay their eggs? Anyway, there are two Rainbow Dragon types (black rainbow and white rainbow) that have a really small chance to hatch from any egg. They can be difficult to get.

Adult Dragons can be mounted like many other animals if you put them in the "pet slot" of your inventory. This makes the Dragon appear in the world and follow the player. Approaching your Dragon and pressing E will allow the player to mount the Dragon.
In addition to having a slightly faster walking speed and a higher jump, a mounted Dragon has the ability to fly. This flight is limited and uses stamina (not the yellow energy bar, but the green circle that appears when you swim). Here are some notes on Dragon flight:

-Dragon flight is activated when the Dragon jumps and the space key is pressed midair.
-When space is held down, the Dragon gains height.
-The glider sound may play when flight is activated even if you don't have a glider.
-If you do have a glider, it will open up when the Dragon starts flying.
-A green circle will appear on the screen while the Dragon is flying. It looks identical to the swim circle. It works in a similar way to the swim circle but it's more forgiving.
-If the flight circle is depleted and you land in water, the screen fades to black as if you had depleted the swim circle. You re-appear in the place you were before you started flying.

I have found a few Dragon-pattern-related bugs while messing around in Creative mode and playing with others:
-Dragons in other players' "pet slots" will look like a Dragon in yours unless your pet slot is empty when the other player joins the world/puts the Dragon in the slot. Baby Dragons are affected but will not appear as adults.
-Dragons in other players' "pet slots" do not appear as the Dragon's real pattern on your end. They will look like the Dragon's type's "base pattern".
-Dragons taken from the Creative inventory are buggy sometimes. Specifically, Spring Dragons put in habitats or the "pet slot" will look like Fairy Dragons, and Fairy Dragons look like Dark Dragons. There is an animal called just "Dragon" that can be taken from the Creative inventory. If I remember correctly it looked like a Black Rainbow Dragon.
These bugs might only be on my end. I don't know if other players have encountered these bugs yet.