Rome is one city that I consider highly underrated, with its attractions seemingly overblown but really quite fantastic. Vatican City and its treasures alone are worth the visit, and the Piazzas all around the city are unique and splendid. A day trip out to Pompeii and the Amalfi Coast are definitely worth it, although it is quite a quick trip!
I can absolutely see why Rome is the capital of Italy - it has the most diverse, interesting and beautiful cultural destinations. Vatican City, with the wondrous Vatican museum and St. Peter's Basilica, is worth every second. The treasures in the museum, culminating in the Sistine Chapel's elaborate ceiling, are truly unique. Along this same vein, St. Peter's is one of the most lavish buildings I have ever seen.
Outside of Vatican city, there are plenty of wonderful places to visit. One of my favorite areas was the Piazza Navona, with the gorgeous Dhiesa di Sant'Agnese in Agone. The sculptures and reliefs in this small church are so skillfully done that they evoke a sense of action and motion that is spectacular. The Fountain of the Four Gods anchors the various merchants in the plaza.
Generally, the typical tourist destinations were not of too much interest. Of course, visiting the Colosseum was necessary, and the Roman Forum was a nice walk. But, I personally prefered the underrated Baths of Caracalla, with huge ceilings and arches that we would be hard pressed to find in most places here in the United States.
As for food, Duecentegrandi, right outside of Vatican City, is one of the most fresh spots.
Pompeii and the Amalfi Coast
The Amalfi Coast was also mostly a disappointment. Positano was my favorite city of the bunch, but it was hard to have enough time to really visit while coming from Rome in a day. The ferry visit over from Positano to Amalfi was very nice. I'd highly recommend seeing the coast from a boat.
Pompeii was one of the most fun places to go. It's incredible how much has been saved over the years despite the weathering effects of sun, wind, and general weather. Vesuvius is due for another eruption "soon", so there is always the chance for lava to once again keep the conditions of Pompeii intact.