Contrary to what you might hear, there is a lot of free music available on the Internet. A good place to start is with podcasts. There are a few different licenses available to podcasters, such as Creative Commons. Podcasters are often also granted rights to play music from smaller labels. In some cases, because the podcaster is not in the U.S., they can also play artists that would never be approved for play within the U.S. Here is three to get you started:
Pandora or Blogmusik. With Pandora, you can suggest an artist or sing that you like, and Pandora will play it. From there, Pandora will play other songs that they believe you will like based on your selection and others before you. This is a great way to discover new songs and new artists. With Blogmusik, you can search on artist or song, and it will return sources for you to select and listen to. Both services play the music through Flash, making it difficult to copy unless you capture the output of your speakers.
Perhaps you have an old vinyl collection, but do not have the tools to convert them to digital. With ourTunes, find others who have the same music on iTunes, and you can copy the files across the network.
If that is not enough music, try Podbop. Podbop helps you find live music in your area, plus if the artist has made it available, you can download tracks from the artists off of the Podbop site.
Of course there are many more music options on the Internet, some free, some not. Using one or more of these options, you can get access to artists from local to international for no cost and DRM free.