Yes, you certainly can!
Use either a King profile or Facebook to easily switch between devices and access your game progress and Gold Bars wherever you play. Boosters and extra lives however, are always stored locally on the device you received them on, so they won't move across devices.
There is a King profile option in your settings menu. Look out for the 'Sign-up with email' prompt, and you can keep your social media separate from your gaming, but still benefit from thousands of other King players who can send you lives.
- Open your game on the old device and choose the 'Sign up with email' option.
- Now choose an avatar and nickname.
- Then (and don't skip this part!) enter your email address, choose a password and that's it – you now have a King account.
- Open the game on your new device, choose the 'Already have a King account/login with email' using the email address and password you just saved. Now you're good to go!