Instructions
Open the city map and go to Magic Town. Select Vicki the Vampiress's lot. Speak with her.
Purchase a Dragon's Nest from Vicki. It costs 49 MagiCoins and comes with a free dragon egg. If you don't have enough MagiCoins, you can earn them by attempting magic tricks while in Magic Town. You don't need to succeed to get the MagiCoins.
Return to your home. Your Sim will automatically drop the Dragon's Nest and egg. If you want to move it to a more convenient location, open Buy Mode and select the nest. Move it like you would any piece of furniture.
Click on the Dragon's Nest. You'll interact with the egg. The more you interact with it, the better the dragon's temperament will be once it's hatched. A dragon with a good temperament is less likely to burn down your house.
Wait a day for the dragon egg to hatch. It emerges full grown and its personality is unchanging. For the most part, you don't need to tend to it to keep it happy, as unlike a Sim, a dragon has few needs. It does, however, need to eat.
Open Buy Mode and purchase flowers. Plant them throughout your yard. The dragon's preferred meal is a flower, and it will eat three to six a day. Restock your flower supply when it starts to run low.
Interact with your dragon when its Social or Fun meters begin to drop. Alternatively, purchase a Dragon Chew Toy from Vicki and give it to your dragon. With a chew toy, your dragon won't need any attention from you.