To have friends (or neighbours) in the game, you will need to connect your game to Facebook.
You will see a Facebook Button on the screen that will guide you through the process of connecting your game.

Any of your friends from Facebook that play Dragon City will be in your game.
This also ensures your game is saved, so if you ever need to change to a newer device, all your progress will be there waiting for you.

If you DO NOT HAVE Facebook, unfortunately this is currently the only way to get them in your game, so maybe consider setting up an account :)