So you bought a new, or new to you device, congratulations! Now you want to transfer all your games and data.
First, rest assured that all purchases you've made will be remembered and always available, on Android (Google Play), Apple (iOS, iPhone) and Amazon (Kindle Fire) devices (although no platform allows you to use purchases across platforms unfortunately).
Apple:
With Apple, if you are transferring your data directly phone to phone, through the Apple system, things "should just work".
Otherwise, if you're installing each app individually, then as long as you haven't disabled iCloud for the app and keep the same AppleID, you will be able to download all your purchases and game history just by installing the app and opening it.
You can check your purchase history so you know exactly what purchases you should have. See: https://support.apple.com/en-us/HT204088. If you have multiple AppleIDs with split purchases, see this consolidation guide: https://support.apple.com/en-us/117294
Google:
Google provides a backup restore that should move all your apps and progress - not just ours. There’s an article on it here: https://support.google.com/android/answer/2819582?hl=en.
Further, the Star Battles app has syncing that allows you to share progress between multiple devices. Tap the Play count at the top left and you'll see an option to connect via your Google account. You were likely prompted to do this when you first installed Star battles and will be again when you install it on the new device. This restores your purchases and game play history and keeps it up to date going forward. (We plan to add this functionality to our other apps over time.)