In Dragon City your mission as the Dragon Master is to breed / train and collect as many Dragons as possible!
Once you have started your adventure, you will find that your first Island gets filled up quite quickly with the Dragons you have found.

So you will need to expand and purchase new islands for more space for Buildings / Habitats and Decorations.
In order to unlock a new Island, you will need to have all of the expansions for the current Island you are on.

To find this information, you can do the following:
1) Tap on the Shop Icon in the bottom right
2) Tap on “Expand”

OR

1) See if your Island has a “For Sale” sign on it
2) Tap on this sign to take you directly to the Expand Menu

You may find Rocks, Trees and Bushes in the way, that stop you from placing your new Buildings.
You will need to remove these to place them and this is really simple:

1) Tap on the obstacle
2) Make sure you have enough Gold to clear them
3) Tap on clear
4) Wait for the timer OR speed them up with Gems