1st stop: Band's website
Always a great place to start. Find out more about them, check the media or songs or albums section for possible mp3s or other fun goodies, and check the news sections that will sometimes have links to other websites that might have music.
2nd stop: Band's Record Label Website
Not many people try this but either the band's website or a quick wiki search will pull up all record labels they have worked with. Lots of labels will offer free music from their bands as a promotional tool. One of the general rules usually is the bigger the label, the less likely to find free music. If they are owned by a company like Sony or Warner or something similar, move on.
3rd Stop: Purevolume.com
Probably my favorite site for music discoveries! Tons of bands have profiles through purevolume and a lot offer free music as incentives. The site even has a chart feature for bands offering downloads (signed and unsigned). Word of caution here though because sometimes fans make profiles and upload music for download that is completely illegal. Tips to find the real deal: if labeled "Pro" at the top (if they are professional), labeled "On Tour", has an actual band description or a decent picture, links to the real band site or itunes. These are definitely safer bets but it can still be hard to tell...
4th stop: Band's Myspace or Facebook
Myspace has closed its mp3s for download but that doesn't mean that some artists won't still find ways around that. A lot of bands will offer special promotions or pre-releases to fans through myspace or facebook and occasionally even twitter so consider adding them.
5th stop: Various other music hosting sites
There are tons of other sites out there like Shockhound, Jamendo, Stereogum, Amazon's free mp3 section, LastFM, Soundclick, and others. Some of these sites might require membership. I'll update as I find and remember more...
6th stop: Google search
The reason this is so far down the list is the temptations are too great and the searches usually too broad. Warning!!! Do not open any sites with mp3 in the name or anything involving torrents. These are mostly completely illegal!! Because google has modifiers be sure to put a - in front of any word that you don't want popping up. I would suggest adding it with the words lyrics and torrent.
Enjoy these tips to finding your own music. If you find anything good, tell me.