Eggs found by Toothless on a search need to be hatched in the Hatchery before the dragon can be placed.

After feeding a Dragon to level up it has to be trained in the Academy to continue to gain levels.

Valka's Sanctuary is where you can train your Dragon to become a Titan. The Titan Training works on chance and the success rate will vary depending on the rarity of that particular Dragon being trained at the time.