How do I carry over my shelter to a new device without losing my progress?

Go to "My Files" on your android launcher and then go to all files -> device storage -> android -> data -> com.gamesas.falloutshelter -> files and then copy the vault*.sav files to memory card or whatever. Then transfer to new device having the game installed and copy file(s) back to same place. Also worth taking periodical backups in case savegame gets corrupted.

Nice one! Thanks for the help!