For Android:

First, log in with your Google account. You can sign in on the main title screen or on the Event Details screen before a race.  

Once you're signed in, your game will start to be synced to the cloud. Backups tend to happen at save points, so make sure you play through at least one race in with an active connection to trigger the sync. When you start the game on a new device with cloud syncing enabled, you should see a message box telling you of a save game conflict, and asking you to pick which game you want to keep. Typically you should choose the game with more progress (on a new device, this should be the cloud game.)

For iOS:

Under the Settings menu, go to iCloud, then select Documents and Data. Make sure it's turned to "ON". Under the Documents and Data menu, select on (or green) for your desired game and you should be good to go.

Backups tend to happen at save points, so make sure you play through at least one race in with an active connection to trigger the sync. When you start the game on a new device with cloud syncing enabled, you should see a message box telling you of a save game conflict, and asking you to pick which game you want to keep.  Typically you should choose the game with more progress (on a new device, this should be the cloud game.)

For Windows:

 
For Riptide GP2, Shine Runner and Beach Buggy Racing, the cloud saving is tied to your Live ID.  Assuming you are logged into Windows with a Live account (and not just a local user account), your game will be automatically saved to the cloud.
 
For Amazon:

Unfortunately, Amazon devices only save locally.