You can't really sync your progress on Android, but you can transfer them using an app like Helium. Follow its instructions to activate the app, then create a backup, transfer it to another device and restore.
As for the game files, I believe it's located in
sdcard\Android\data\com.budgestudios.googleplay.MyLittlePonyPocketPonies (with "sdcard" being the location you installed the game).