Go to the Enchanted Valley and talk to Fairy Mymyu. She will tell you about the dragons that are scattered all over the world. We need to talk to one of them (exactly one, we don't need to rush around all of them), and bring the egg of this dragon. From the egg, the fairy will make us a dragon.
Quest that must be completed in order to obtain a pet hatchling. Visit Pet Manager Cooper and tell him that you wish to raise a hatchling. He will tell you to seek help from Wiseman Cronos of Hunters Village.
Make a Fairy Stone
After collecting the items requested by Wiseman Cronos, find Maria in Dion Castle Town and ask her to make a fairy stone for you. Take the fairy stone to Cronos and he'll tell you what you need to do next.
Fairy's Existence
Take the completed Elven stone to Guard Byron and ask him about fairies.
In Search of a Fairy
Go to Fairy Valley and find a fairy who will talk with you.
In Search of a Dragon Species
Take the juice of monkshood that Fairy Mymyu gave you to the drake or wyrm that she introduced you to.