When I was on iOS I used gMusic for that and it worked pretty well.
Considering Google puts so many of their own apps on iOS, I'm surprised they haven't made an official Google Music app for iOS. But gMusic got the job done just fine when I used it.
Does it allow you to download music locally to your device for offline listening? And search for new songs from Google? It wasn't the apps that drove me from iOS to Android though even if they released an app for iOS I wouldn't switch. I'm betting it has something to do with Apple taking a 30% cut of Google Music subscriptions.