How can I move my game to my new phone?

You need to bind the game first, if you haven't done it already.

Go to the Settings menu and the Account option and choose to bind to Facebook or Google play.

Once done you can load the game on any other device running the game. Install the game on your new phone. You will need to play through the tutorial section of the game until you can access the settings menu again. This is around chapter 3. Go to the account page and choose the Switch account option. Log in with the same external account to load the old game data.