In order for your Dragons to flourish, you will need to build the correct habitats for them to live in. All of these can be purchased in the build menu and constructed by clicking on a free space in your kingdom.
The Build Menus is where you should go to buy everything you need in the game, including Eggs, Habitats, Buildings, Decorations, Expansions and Treasture. You can also use it as a place to store items you no longer need.
Initially, you don't have too many Dragons at your disposal. Other Dragons are unlocked as you progress through the game - most need higher levels before you can get their eggs.
The Deus Vault is a building which allows you to purchase formulae for breeding certain types of Dragons. It costs 500,000 Gold to purchase, and this can only be done once you have reached Level 20. Once you have purchased a particular formula, in then becomes possible to breed that particular Dragon.