For Android users:

On your original device, open the game and click the settings button (gear icon) in the upper-right corner of the screen. Go to the Community page and link your Google account. Then, on your new device, log in to the same Google account to sync your progress.

For iOS users:

On your original device, open the game and click the settings button (gear icon) in the upper-right corner of the screen. Navigate to the Community page and link your Game Center account. Then, on your new device, log in to the same Game Center account to sync your progress.