Titanium Backup (or
Pro, and IMO worth it) on the Google Play store.
There is bad news though: if you aren't rooted already, then it will wipe your device when rooting, meaning all your progress is gone anyways, unless it stores it somewhere OTHER than the app's private directories.
For example, the Square-Enix games (Final Fantasy Dimensions, IV, etc) store it in the private root-only directory of "/data/data/com.sqaure_enix.android_googleplay.ffl_gp/files" (this is FF Dimensions). It's that second "data" nesting that's visible to the app itself, and root only. Get the files in there to keep your backup across phone wipes.
But Minecraft Pocket Edition saves its files in "/storage/sdcard0/games/com.mojang/" (sometimes the first part is "/storage/emulated/0/" and not "/sdcard0"). This means that you can always access it with non-root and copy those files around as you need to. One downside here though is that things like Titanium Backup don't "know" about these by default, so you may need to manually back these up, or set up a rule somewhere for your backup utility to grab it.
So to get everything, get Titanium Backup. But check first to see if specific apps are either storing them on the cloud, or are storing them someplace else. A good way to tell if cloud is if you need to login to them, AND you can continue your game on any of your devices (phone AND tablet, and it's the same game).
Either way, always root first. Then you won't need to wipe it just to install a backup utility.