Do:  Go ahead and take an all day tour of Ancient and Monumental Rome.  You could perhaps get away with doing it cheaper on your own, but the convenience and comprehensive nature of the tour make it worthwhile.

Do:  Go to Pompeii for the day.  Consider the Vastours day trip to Pompeii/Naples, even though the lunch is below average and the drinks in the restaurant are outrageously priced.  Also, if you wish to actually see Naples, the trip does not really offer such an experience.  But a visit to Pompeii is worth it.  For experienced travelers, you may wish to bypass a guided tour and take the train from Rome down. 

Do:  Taking the train yourself up to Florence for the day if you are only staying in Rome, or consider paying for an all-day tour.

Do:  Pick a restaurant in the Parthenon square area for dinner and then walk to the Trevi Fountain at night.  On your way, stop off at Gelatti's for ice cream.

Do:  Eat at Brasserie's Restaurant (Pizza) in Vatican City near the Hotel Delle Vittorie.

Do: Read the book "As the Romans Do" by Alan Epstein before you go.  Consider booking the author for a day or two for a private tour while you are in Rome; he lives there. 

Avoid:  Unless you are a devoted Catholic, suggest you to go to the Vatican on your own.  Purchase tickets online through the Vatican's site and save on paying for a guided tour, which is longer than need be and a bit tedious.