Visit a lyric search engine (see resources). Enter as much information about the song as possible, including lyrics, artist and title, along with album information, and full lyrics will appear. This free tool can be used to track down songs so old that only a few lines remain in the listener's head.
Visit a vintage or used-music store. These stores stock many old albums by artists not featured in mainstream music stores and often catalog them.
Search for the song on media-sharing sites like YouTube, GrooveShark or Spotify. Music can be heard for free on such services, which feature simple interfaces and can often be accessed from mobile devices.
Purchase the old music online. There is nothing quite like having the original vinyl LP by a favorite artist -- and a number of sites (see references) sell them. Sites like eBay and Amazon may also list old albums for sale.