What chords to use is definately a lifelong study. The method already suggested of starting with a melody is definately a good one. One thng I like to do is find a nice melody, then a nice, melodic bass line, and then see what kinds of chords will work with that to make a strong, interesting, musically logical pattern. If you keep in mind that the two notes you're working with can each be the root, the third, the fifth, the seventh or even a more distant extention, you really have more possibilities than you can use. (Of course not all choices work equally well. Often a third, seventh, ninth etc. can sound fairly dissonant in the bass. By contrast, these may be the most beautiful choices you have in the melody.) Just using major and minor chords gives you a huge range of possibilities, and adding dimin., half-dimin., augmented, sus, etc gives you even more choices. I often try to keep root motion (as opposed to the actual bass line) to moving mostly in fourths, fifths, and major and minor seconds. This is by no means a rule, but it does help to lend some structure and help out the listener to make sense of the song.
Breaking down songs you like--like standards, Beatles songs,or Dave Matthew's or whatever--can really help you to understand what you like and why it sounds the way it does. See what kind of progressions you like and try to write some songs based on them. Harmony is a really fun and endless part of musical learning. I read about Paul Simon taking some harmony theory classes in the mid-seventies. This would be years after already writing all those Simon and Garfunkel songs. If P. S. can use lessons and still has much to learn, think about how much cool stuff there is out there! Have Fun.
Peace,
Matthew
PS Another great thing to do is just sit down with your intrument and sing. Then you know it will be real, and not just head music.