Apart from the subway to the Vatican, I didn't use any buses or trams to get around Rome because they were completely full to overflowing and looked a bit dangerous. Many of the sites in Rome are all over the place as well and take a long time to walk between.

I definitely think that the Vatican is the best place to go for one day. Great views, fairly safe once you're in, compact and easy to get to and from. Just watch the beggars outside.

Another alternative is the Roman forum, but it can be quite crowded during the day.

The train from FCO takes about 45 minutes IIRC so you may want to factor that in to your available time as well. It gets pretty full too, so put your baggage directly above your seat where you can see it. Return to the airport with plenty of time to spare because Trenitalia is probably the worst train system I have ever caught (massive delays and cancellations without explanation).