Fallout Shelter is now out for Android devices and for some reason there is no way to sync your game account with other devices such as tab/phones or PC. So to take a backup and transfer you save game to another device here is the complete guide explains Step by Step how to transfer.
If in case you delete or the app or update your phone without this backup your save game will be lost with the purchases you have done in-game(in game expenses and real money) and you will have to start all over from the scratch. So to avoid such situations you should always have this backup done before any changes to your phone or memory.
For all the android devices if you also try copy/paste the save game between the devices it cannot perform the live sync as it does for other games. But if you have spent real money in the game then you should not take the risk. The save game in located at the below location:
/storage/sdcard/android/data/com.bethsoft.falloutshelter/files/VaultX.sav
In VaultX, the X represents the vault you want to take backup. For Example Vault1/2/3(save vaults).
You can use any File manager(we recommend "ES File Explorer File Manager") to navigate to that folder. After you open the folder you will find ".sav" file you can upload it to your Dropbox and it's done. Your save game is now backed up and safe. If you want to move this save file into another device then you can copy this files from Dropbox and paste it to the location mentioned above.
Note: It might overwrite your current vault.
Don't try to edit the .sav file for fun and profit as it might damage your current progress. Enjoy and play safe.