Rick's mention of harmony singing with a partner reminded me of a tip. If you don't have a partner, but want to learn to sing harmony, a great place to start is by listening to the Everley Brothers. First listen to what they did, then try singing along with one part, then the other, and get comfortable with how they fit together. Then when you're ready for the next step, try adding a third vocal line of your own. The songs are "standard" enough that it is generally apparent what is "right," but they're inventive enough that it's not a cakewalk. Start with the easy ones -- Bye Bye Love, etc. -- then move on to Devoted to you, Crying In The Rain, etc. Guaranteed to develop your ear, and your singing prowess.
There are other people out there that may be equally valid to do this with, but it's hard to go wrong with the Everleys -- and the fact that there's only two of them means that there's always room for a third line.