take a train from fco to termini (the main train station in Rome). I think it runs every 15-30 minutes, so you shouldn't have a problem catching one. Once you get to termini, you can take a bus to where ever you want to go. There's really not much to see around the train station, so you'll want to get going pretty quickly. Bus 64 goes down Corso Vittorio Emanuele, which will link you to Rome's "hot spots" (the forum, the type-writer, piazza navona, and the vatican), but you'll need to watch out for pick pockets (bus 64 is also known as the pick-pocket express).

A few other ideas:
St. Peter's is worth it (be sure to stand at one of the "dots" in the pavement at Bernini's arms and watch the columns line up!), but the Vatican Museums are not if you're only there for a day. The sistene chapel is beautiful, but the long walk through the museum to get there isn't a good idea on a short time frame.
Visit Trastevere for good restaurants and an authentic "old rome" feel - it's across the ponte sisto.
you MUST go to the pantheon (though not the mcdonalds across from it)! if you're lucky it'll be raining and you can stand in the middle. You can get there easily from Piazza Navona.

PM me if you want more specifics - I lived there for several months (and not too long ago!)

Have a great trip!