Rome rewards those who understand that the whole city is a museum. The Colosseum (pre-book tickets — the queue without is hours long), Roman Forum, and Palatine Hill are covered by a combined ticket and deserve at least half a day. The Vatican is a full day: Vatican Museums (Raphael Rooms, Gallery of Maps, Sistine Chapel) and St. Peter's Basilica — book the dome climb separately for the best Rome panorama.
Beyond the blockbusters: the Pantheon (free, arrive early), Piazza Navona's Baroque fountains, Campo de' Fiori market in the morning and aperitivo spot at night, Trastevere's narrow streets, and the Appian Way by bike on Sundays when it's car-free. Throw a coin in the Trevi Fountain, but come after midnight when crowds thin.