To transfer your progress to another Android/iOS device:
1. Make sure that your in-game profile is connected to your Google Play or Game Center account
2. After that, enter the game with the same Google Play or Game Center account on your other device
3. If, after starting the game, you aren’t offered to restore saved progress, go into the in-game settings and tap "Connect"
Note that you can't transfer progress from Android to iOS and vice versa.