We visited Rome for 3 days in Fall, 2010. Rome is a dream come true for people interested in ancient history, monuments, churches and palaces. It is like an entire civilization in a city. You can just keep walking and you are likely to encounter one impressive monument after another, e.g. Piazza Venezia in the pic below.
Even for people like us, who usually avoid visiting museums / churches, the monuments here are really impressive and you should plan for at least a 5 day trip to fully explore the city.
The downside is that Rome is almost always teeming with tourists, so expect long queues at touristic places. Hotels are expensive but food / public transportation are still quite cheap (as compared to say Paris). You will also find people selling stuff on the streets here, not only souvenirs, but also products like shoes, handbags, bangles, clothes, etc. This was definitely a novelty for us, as compared to other European cities. If you are one of those Indian tourists, who go looking for an Indian restaurant wherever they go, then fret not as there are plenty of Indian restaurants here. If you are the more adventerous type, and prefer sampling the local food, then there is of course no shortage of Pizzerias here. If you are vegetarian, try the Pizza Margherita - you will find it in almost every Pizzeria and it is definitely vegetarian.
Vatican city is a country in itself, and is probably one of the smallest in the world. The good thing is that you do not need a separate VISA or any additional documents to enter Vatican city. It is probably also a good idea to invest 50 Euros in a tour guide as they know the right entrances - avoiding the long queues sometimes - and also the city has a lot of things to explore in a day. You will probably not get lost but a guide is definitely recommended to make the most of your trip.
For the uninitiated, this is of course the place where the Pope stays. You are unlikely to see / meet the Pope unless you visit the Vatican on special (festival) days. The best you can hope for is to see the stairs leading up to his chambers (pic below).
Stepping outside, one can see the huge Saint Peter's Square - which you might have seen on TV whenever the Pope makes important announcements.
St. Peter's Basilica is an equally impressive church in the vicinity. Inside, you can find a number of masterpieces, each rivaling the other in beauty, e.g. Pietà by Michelangelo (pic below).
Everyone who has seen movies such as Ben-Hur or Gladiator, or is aware of the legend of gladiators, would be aware of the Colosseum in Rome. The structure is huge and although some parts of it have been destroyed, one can clearly see the prisons where gladiators / animals were kept, and how the whole proceedings unfolded.
There is also a night tour that takes you much further into the Colosseum, than is usually allowed in the day, and also in smaller groups (of max. 25). However, the tours only run on specific days and they get booked months in advance. So be sure to plan for it in advance. You can get more details here.
There is also a lights display during evenings / nights that simulates the Colosseum burning. This is free, happens every night, and is quite impressive (pic below). So, don't miss it.
Trevi Fountain (Fontana di Trevi)
Trevi is one of the largest and most famous fountains in the world. Try visiting it in the night and enjoy the atmosphere. The good thing about Rome is that almost all the touristic places are easily reachable by public transportation and are well marked.
Culture / Music
Rome was not only the political, but also the cultural, foci of an entire civilization. There are numerous magnificent auditoriums where you can enjoy and relive the music / opera of the bygone era. One of them is the newly constructed Auditorium Parco Della Musica where we had the pleasure of attending a Beethoven concert (Synphonies: N. 6 "Pastorale" and N. 7) by Kurt Masur. The performance was an extremely enriching experience, and we would highly recommend it even if (like us) you have never been exposed to western classical performances before.