Of course! Open the game on your old device, log in to Facebook and play any Single Player level. After that, simply log in with Facebook on your new device to recover your progress.
Purchased bikes can be restored by downloading the game with the same Apple/Google ID used to make the purchases and tapping on Restore in the game's Shop. Please note that purchased bikes will only be restored if they were bought on the same game (Bike Race Free/Bike Race Pro), store (App Store/Play Store), and account (Apple ID/Google ID). This means that if you purchased a bike on an Android device, you will not be able to recover it on an Apple device. This also means that you will not be able to recover bikes bought on Bike Race Free in Bike Race Pro.