In order to expand your Park, you will need to unlock new areas. This can be done by tapping on the locked areas. You may need to gain some additional levels to unlock areas further from the starting point. These areas will have an icon displaying a lock and the level required, while areas that can be unlocked will show a different icon.
When tapping on the area you wish to unlock, you will see that you need to provide specific items as well as currency. These required items can vary.
New zones might expand your population limit, give you a new Ride or a new Cooking building. It's recommended to unlock the Cooking building zones as they will provide you with new items and recipes.