Please note that once you connect the game to Google Play, you won't be able to disconnect the game from it or use this Google Play account to connect another game to it.
You can connect the game to your Google account to save the progress. There are two possible situations:
Connecting upon the first visit
When you enter the game for the first time, you will be asked to save the game with Google Play:
Give all the permissions required. There you go, your game is now safe and sound on your Google account!
Connecting when necessary
If you don't want to connect the game at the moment, you can always go to the game Settings and tap on the Google Play button to connect the game to it.
Progress will be saved on your Google account.
Changing Google Play account
Tap on the Google Play button to change accounts.
The game will reload and you will be able to connect it to other Google account.