1: Elder Cronos
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 Elder Cronos of Hunters Village.
2: Make a Fairy Stone
After collecting the items requested by Elder Cronos, find Maria in the Town of Dion 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.
3: Fairy's Existence
Take the completed Elven stone to Guard Byron and ask him about fairies.
4: In Search of a Fairy
Go to Fairy Valley and find a fairy who will talk with you.
5: In Search of a Dragon Species
Take the juice of monkshood that Fairy Mimyu gave you to the drake or wyrm that she introduced you to.
6: Find the Stolen Eggs
Recover 20 stolen eggs.
7: Hatching Ritual
Take the eggs to Fairy Mimyu.