So, starting with your Dragons is pretty easy, if you are buying a starter dragon from Mystical Creatures or an LE you can just birth the Dragons however if they are Dragons, you bred or brought from someone else who was breeding them then you will first need to incubate them before you can birth them. You will need a Mystical Dragons Incubator:
These are free from any Mystical Creatures store. To use them you would just click on your Dragon egg that you want to eventually birth and click on incubate. Make sure you do have an incubator close.
You can have as many incubators as you want on your SIM.