Nothing on the iPad offers as much flexibility (both free and paid) as the Music app and iTunes. As long as you own a computer you can rip your music from other formats (that you own) for free. You can listen to the iTunes Radio (iOS 7) for free (or pay if you don't like the few adds). Or you can buy music from one of the largest online catalogs in existence.
{availability of iTunes Radio is currently limited to the U.S., and I believe Australia}
That's the best genreral-all-arround solution, in my opinion.
If you have more specific requirements, like best streaming, best availability on multiple platforms, radio stations, cheapest, etc., the answer is likely to be something else. There is no best at everything app or service, free or paid.
On the other hand, you are hardly limited to just one app or service. While you probably want to pick one place to pay for content (most of it anywya), you can has as many sources of streaming content as you like. Most offer both free and paid models, so you can try and decide what you like best for yourself.
Though I seldom use them these days I have the Pandora, TuneIn Radio, and NPR Music apps. Useful when I want a change of pace from iTunes and the Music app. I also keep OPlayer HD on my iPad for those occasional files that are not compatible with iTunes, and are not worth converting and importing.
And there is YouTube, of course. Lots of music there. I find it handy, mostly, for sharing an example of an artist I like with others. Those others mostly being here on this forum.