The best way to save your game progress is to connect your game to another service. To do so, from the game's main menu tap the cog icon at the top right of the screen, then tap About, then Connect. Here you will be able to connect to Facebook if you are using an Android device, or connect to Game Center if you are using an Apple device. Connecting will allow you to restore your game onto any compatible device logged into your Facebook or Game Center account respectively.
It is also a good idea to record your User ID. Your User ID can be located in the settings menu below the social media buttons. You may also be able to locate your User ID by taking a screenshot while the game is initially loading (your ID is at the bottom). This ID may help the support team to restore your account if it is lost or if you want to switch from an Apple device to an Android device or vice versa.