Mail the lyrics or a recording of the song to yourself. Leave the package unopened when you receive it. This is a simple and free way of retaining proof of your work. The date on the package will provide an official date, verifying that the material inside is your own. However, this will not allow you to collect on royalties or any of the other perks of a published songwriter--it is merely a way of proving your legal ownership.
Register with ASCAP. ASCAP (the American Society of Composers, Authors and Publishers) is an organization designed to protect the rights of a songwriter (performer, etc.). There is a one-time registration fee, but once you are registered, you are registered for life. On becoming a member, your songs are registered with ASCAP, which then collects and pays out royalties for performances, airplay, or other recording use of your songs.
Register with BMI. BMI (Broadcast Music, Inc.) is the largest alternative to ASCAP and provides many of the same services. BMI is free for songwriters, though there is a fee for publishers and classical composers. Like ASCAP, once registered, BMI collects licensing fees when your song is played on air, performed live, or recorded by another artist. Royalties are payed out quarterly.
Copyright your work. You can register copyright of your songs with the U.S. Copyright Office (www.copyright.gov). This will go a long way in establishing your ownership of your creative work and your negotiating leverage when potentially dealing with a record label down the line. If you choose to forgo groups such as ASCAP and BMI and simply register with the copyright office, then you may have to be on the lookout for performances, covers, or recordings of your material by other artists.