Instructions
Get the dragon amulet and increase your character to level 10. You should already have the amulet from growing the baby dragon.
Go to Lady Celestia in Sunbreeze Grove. You should have already spoken with her several times in your quest to get a dragon. Speak to her.
Click on "Dragon Quests." Select "Summon." This summons your adult dragon for training.
Play the "Togzilla Attacks!" quest. This quest is repeatable and gets more difficult each time it's played. Every time you beat the quest, your dragon learns new skills and abilities.
Click on "Dragonlord" and select "Dragon Training." You'll need to have summoned your dragon to start training.
Train your dragon further by battling staff dragons in the Dragon Arena. This is under the option titled "Dragonlord." You must reach level 20 before you can access the arena.