How to transfer save data from one device to another?

[AC Question]

I have a Sony Z5 full of games, how do I transfer all its save data to my new Samsung S7 when I finally get it?

iOS has iTunes/iCloud back ups, so never had problems transferring games and its save data across new iOS devices but with Android it has left me scratching my head.

I haven't got a rooted device and I don't plan to either, I know there are some backup apps but the ones I downloaded only backup the .apk file which is pointless as I can just re-download stuff from the play store.

Thanks