Check out the legal free music from established music download sites. Many big online music retailers like iTunes, MP3.com, Amazon,com, and more have free music sections. You can legally download these free tracks from some artists that you like or some new ones that you would like to listen to for the first time.
Check out mainstream MP3 stores and look for their free music section. Often this is populated with songs from up and coming artists which are worth checking out. For example, MP3.com has a whole category dedicated to free songs. You can sample each and then download those that sound great.
For free music without actually downloading, you can sign up for online radio. You can "like" certain artists and songs to customize a station, so you only get the music you want! While Pandora is highest in popularity, a couple of other options with similar quality are Slacker and Last.fm.
Do not download music illegally through file-sharing websites. What seems like a good ty segall idea to get free music can come back to bite you. Many internet service providers are now working with publishing companies to find and prosecute online pirates, and you do not want to be one of the unlucky ones.
Make sure you have broadband internet connection if you download music online. If you have a dial-up connection, the download will be much slower and the files will take forever to transfer. So if you want to enjoy your music right away, a broadband connection is the way to go.