You get new islands by leveling up. When a new island comes available, you will need to purchase it with coins. This means one of the best methods is to breed and sell new dragons. This will get you both XP and the coins you will eventually need.
You don't really beat levels in DragonVale. You just get enough experience points to pass to the next one. You get experience points by clearing rocks and trees, buying decorations, breeding new dragons, and a few other actions.