To save and transfer your game progress between devices you will first need to make sure that your game progress has been saved to the cloud.
To do this, sure that you are signed into either Facebook or your Apple ID (for iPhones/iPad users) in the game. This can be accessed via the game's Setting menu (cog icon on the top right). Click either the Sign-in with Facebook or Sign-in with Apple link.
(currently, Google Play sign-in is not available but we will be adding it in the future)
So that no progress is lost, always ensure your game stays signed into the required system.
Download and install the game on your new device and follow the same steps to either sign into Facebook or Apple in the game. This will then prompt and ask if you would like to load a cloud save, select 'Load Progress from...'.
Select 'Yes' if asked to confirm your selection. This will now download your save onto your new device.