For Tap Zoo specifically, you can save your game on their server and then load it onto another device and its really easy!
Here are the instructions to Register to permanently transfer your game:
1) In your game on your old device, Save your game first. Then go to Main Menu > Other > Account > tap on Register and follow the instructions
2) On your new device, download the game (if you haven’t already done so) and do step #1. It should now say that you’ve already registered that email address, and to check your email.
3) In your email, you will receive a short note with a link that you must follow to confirm that you want to link the game between the two phones.
4) Immediately after following the link and seeing confirmation that you have linked the game across devices, you can load your game on your new phone.
5) To load your game on the new device, in the menus find the “Save/Load” button. Then click “Load”
http://pocketgems.com/support