When you sign in with Google Play Games, your Iron Marine's Invasion progress will be saved automatically. This means that you'll be able to pick up where you left off, even if you uninstall the game or switch phones, as long as you use the same account and have a connection to the cloud.
Before you start
Make sure you have Google Play Games installed and updated.
This is not required, but it does help with the sync.
To enable Cloud Save
Open the game and load your save file.
From the map, go to the Settings Menu on the bottom left.
On the top right options, find Cloud Save and check if the option is enabled.
This is how the enabled cloud save looks like.
Each time you launch the game, you'll see a confirmation message from Google Play Games.