You're working hard on your island, decorating it to your liking, completing tasks and it would be very bad to lose it, isn't it? But everything could happen, you can move to the new device or just accidentally delete the game!
To prevent it, please, open Settings. Here you can see two buttons: connect to Facebook and connect to Apple ID.
   
Choose the method that best fits you and press the button. Log in to the corresponding account and follow the instructions on the screen. If the connection was successful, the bottom will display "Connected".

Now your account is secured! From now on, you can log in to your old account from any new device.