Unfortunately, Game Center and Google Play will not save your game's progress since they only record your achievements. Currently, the only way to connect your game to an account to save progress and access it across multiple devices is by using a Jam City ID account. The Jam City ID login can be accessed by tapping the gear button on the upper right corner, which will bring up the Settings menu.

For more information about the login process, please see the "How do I save my game to an account?" and "How do I load a game that was saved to an account?" FAQs.