Register the new device to the same account and you download apps, music etc. Game progress may or may not be saved, that is up to the individual app developer.
If the old one still works, you can backup
Silk Browser bookmarks
Home screen and app layout
Note: Purchased music, movies, TV shows, apps, and books are automatically stored in the Cloud and can be downloaded to your device from the Cloud tab. Photos and personal videos can be backed up in your Amazon Drive. To learn more, go to What Is Amazon Drive."
NOTE: I do NOT speak for or represent Amazon in any fashion. This is a customer to customer forum.
Disclaimer: I have examined any sites I provide links for and have determined them to be safe but I make no promises about safety, content, ads served or anything that might make you clutch your pearls in horror. Use discretion clicking on links provided by total strangers such as myself.