By far the best radio iphone app that is intuitive and easy to use is the TuneIn radio app. You can search for stations, find stations by music genre, find stations by Sports and Teams, find Talk radio stations, or even find stations based on language.
It also suggests all the local radio stations in your area that you can listen to. Furthermore, you can even record radio snippets directly from the app. They have over 40,000 stations to choose from and I've also noticed a lot of internet radio stations that you might have listened to in iTunes radio that can also be heard directly from the TuneIn radio app.
If there's one radio app you want to have, the TuneIn radio app is definitely the one to get and from what I've seen of other radio apps, TuneIn is by far the best radio app for your iPhone, iPod Touch or iPad.
Keep in mind, If you are just interested in listening to music and note actual radio stations, then you'll definitely also want the Pandora app and/ or the Last.fm app.