Rome rewards a plan, because its biggest sights — the Colosseum, the Vatican, the Pantheon — are spread across the city and each deserves unhurried time. This itinerary splits ancient Rome, the Vatican, and the historic center into separate days so you're not racing across town.
Book a timed entry for the Colosseum first thing, which usually includes access to the Roman Forum and Palatine Hill next door — plan at least three hours for all three. In the afternoon, toss a coin at the Trevi Fountain and walk to the Pantheon, still the best-preserved building from ancient Rome. End the day in Piazza Navona for dinner among its street performers and fountains.
Arrive early — the Vatican Museums & Sistine Chapel get crowded fast, and a skip-the-line tour is worth it. Continue to St. Peter's Basilica, then climb the dome for a view over Rome and the Vatican gardens. Spend the afternoon in Trastevere across the river, Rome's most atmospheric neighborhood for an aperitivo and a long dinner.
Climb the Spanish Steps in the morning, then browse Via Condotti for its designer storefronts. Visit the Borghese Gallery (book ahead — entry is timed and limited) and walk through the surrounding Villa Borghese Gardens afterward. If you have a fourth day, Pompeii & Mount Vesuvius makes for a full but worthwhile day trip by train.