How do I save and/or transfer my progress on Android?

You can store your game to the cloud and restore it after resetting or changing your device. 

First, sign in to Google Play Game Services 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 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.)