Can I transfer my Google Play music onto my iPhone? I had a Galaxy S4 and I just bought a iPhone 6. Is there a way I can transfer my music using an app to iTunes? Or do I have to download iTunes onto a computer?
you won't be able to transfer your music directly to iTunes.

You can download this music on your computer per Goolge Music Manager and them sync with your iPhone 
(https://play.google.com/music/listen?u=0#/manager)

Obviously you can use the Google Music apps on your phone too and have the music to play offline too.