Ive been playing for about a week and I realized that I wanted to have a backup in case something happened to my phone. I was wondering if there was a way to actually save the game so that if I lose the data I can use a backup file or something like that.

I will google it for you:

https://www.google.com/search?q=fallout+shelter+save+game+location

I did this but it's for android:

https://www.reddit.com/r/foshelter/comments/3gyjwk/androidguide_how_to_back_up_transfer_your_save/

I recommened you save the folder just to be safe.

The answer to your question Aegis is NO, there is no way to officially save the game or transfer your vault(s). And this should be Bethesda's main objective on their next update-patch.