Saving your account progress will let you play on other Android devices and recover it in case the application is deleted or reinstalled. Therefore, you should do it as soon as possible.
To save your progress you'd need to create a profile on Google Play Games service.

To save your account:
- enter the game settings by tapping/clicking on the cogwheel placed in the upper right corner of the main game screen
- press the "Connect" button

Once done, the progress of your account will be saved on the Google Play Games service.
If the game is reinstalled or played on another device, you'll need to retake all the steps to load your saved profile. The device must also be synchronized with the same Google profile on which your main profile has been saved.
After choosing the "Connect" option, a window will appear. It'll let you choose your main profile. You'll have to choose the account which will be loaded on that device. The other, unselected profile will be lost.