Rome has some of the best Piazzas in the world. My favorite were Piazza del Popolo and Piazza Navona! Piazza del Popolo has the great twin churches. Piazza Navona has the 4 Rivers fountain and is a great person to people watch or eat lunch.
We also visited Piazza di Spagna, but I just found that to overly busy and not have much to do. Piazza di Trevi was also interesting but it was very busy too. You could barely make your way up to the fountain but everybody has to go there when they go to Rome!