How many days should I spend in Rome?

Four is right for a first visit; three is the honest minimum. The Colosseum-Forum-Palatine complex is most of a day, the Vatican Museums and St Peter's is most of another, and the historic centre on foot is a third. That's three days spent entirely on the famous things — the fourth is the one where Rome stops being a checklist: a neighbourhood market, a slow lunch, a train out to the Castelli Romani for wine, an evening walk with gelato. Five days lets you add a day trip to Ostia Antica or Tivoli without cutting anything.