There are so many options to get music onto your devices these days, I suppose it depends what you want to do with it as to which one is best. Obviously iTunes is the most comprehensive, but there is a cost involved. If you are just concerned with listening to music on your phone on the move (and have a substantial data plan) then Spotify and Grooveshark are good options. There is a subscription cost with Spotify to listen ad-free, and make tracks available offline, but the free version works well too.
There are other less known apps available for your phone, but some are only available for Android and vice versa. I would recommend GTunes for Android.
If you have an Apple computer a final option would be to download Audio Hijack Pro. There is a one off cost involved- about USD$30, but once you have it, it allows you to record music from any source and transfer it in to your iTunes to keep forever! So you can listen to a whole album on Spotify, or YouTube, record it, and then transfer to your phone. It's a bit more work, but completely free (after the initial outlay). This is what I use and I love it!