In iOS, podcasts have been relocated from the Music app to the Podcasts app. I think you have to download the Podcasts app seperately (free). Once you're looking at your downloaded podcast in the Podcasts app, you can subscribe to the podcast and get future episodes automatically.
ALSO. the Podcasts app is an uncommonly bad app. Poor navigation, bad performance, frequent crashes. I'm a relatively heavy podcast user and it just didn't work for me. My replacement app for managing and listening to podcasts is
Downcast, right now it's $1.99 and totally worth it.