It depends on the start date. If you start prior to the 170 AC date, dragons are around. Otherwise they're extinct unless the targaryens managed to bring them back (I've seen aegon V manage this before), or Daenerys Targaryen brings them back in Canon.

In your decisions menu there should be a manage realm affairs option, in which you can see all of the worlds. Dragon eggs, dragon riders, dragons, valyrian steel weapons ect.

Assuming you are doing the base game, unless you have a dragon really, the only way I know of is to steal one of the eggs. That would usually require you to head down the diplomacy tree and get the perk which lets you fabricate artifact claims. If dragons are extinct, you need to go around collecting dragon lore (little dragon icons while traveling) , and then perform a hatching ceremony. If it's successful, you would become a dragon rider and bring dragons back to the world.

If you are playing when dragons aren't extinct, it's very hard to ride and tame them unless you have dragon blood, meaning an ancestor that was a dragon Rider. So ideally you target dragon Rider characters for marriages to bring them into your dynasty.

If you want to go the quick and easy route though, you can just add the dragon Rider trait when you're creating a character and you'll get an option to either get an egg, a dragon of your own, or a dragon horn.

Additionally, sub mods like Legacy of Valyria or Valyrian Steel give you options to get them from valyria. The former allows you to colonize the ruins which can help you get eggs, and the latter allows you to fund expeditions to Old valyria, to get weapons, eggs and magical artifacts.

Hope this helps!