If you want to use a Dragon in a Battle, you must train it first from baby to adult.

Access your roster by clicking on the symbol on the bottom left on the home screen. Click on a new Dragon you wish to train.
Before you can train your Dragon again, it needs to be leveled up first. Look at the Training Requirements in the dragon stats to see what level it needs to be trained.

With each completed training, the Dragon will gain a new golden star representing its training tier. How many golden stars a Dragon can reach will depend on its rarity.