I got a new phone and I want to transfer my vault from my old phone to my new one. How would I go about doing that?

You've got two options:

1. Use the cloud save feature. When you're looking at the list of your vaults, put a checkmark in the little cloud box. That stores your vaults on Google's servers. I've never actually used this, so I can't go into details.

2. Move them manually yourself. Use your file manager app(every Android I've seen comes with one) to browse to Phone storage\Android\data\com.gamesas.falloutshelter\files\ and back up all the vault.sav files. Personally I just email them to myself, since it's like a cloud save you can control.