How do I buy islands?

Tap the islands map icon which is at the top right of the game screen, you'll then see the next island which is available to buy. It will also tell you how much it costs.

Once you have enough money check back on that map icon and purchase the new island by tapping it.

The old island will go (be sold), although you cannot take any of the profits to the next island - You have to start over, but with a better profit multiplier, so it's actually really easy to get back to where you were in terms of money really quickly