Rome packs three cities into one — ancient, papal and baroque — and they are geographically separate enough that trying to mix them each day means crossing the centre repeatedly. Take one per day and it works.
Before you go: two bookings
The Colosseum, which uses timed entry and whose ticket normally also admits you to the Roman Forum and Palatine Hill, and the Vatican Museums. Both sell out in peak season, and the Vatican queue without a booking regularly runs to hours.
Note the dress code, because it is enforced at the door of St Peter's and the museums: shoulders and knees covered, for everyone. People are turned away every day in summer.
Day one: ancient Rome
Colosseum at your slot, ideally early. Then the Forum and Palatine on the same ticket — and do them in that order, because the Palatine looks down over the Forum and makes sense of what you have just walked through.
The Forum is largely unshaded and there is very little water. Bring some, and go early or late in summer rather than at midday.
Afternoon: the Capitoline Hill above the Forum, and the view from behind the Palazzo Senatorio, which is one of the best free things in the city.
Day two: the Vatican
Go at opening. The Vatican Museums are a one-way system that ends at the Sistine Chapel, and the difference between arriving first and arriving at eleven is the difference between looking at the ceiling and being carried past it.
Afterwards, St Peter's Basilica. If you have the legs, climb the dome — the interior stair is narrow and leaning near the top, and the view over the square and the city is worth it.
Then walk to Trastevere across the river. It is the right afternoon for it: narrow streets, ivy, and the best evening eating in the city. Santa Maria in Trastevere has mosaics worth ten minutes.
Day three: baroque Rome, on foot
This is the day with no bookings and it is usually the one people remember. A loop: the Pantheon — free, and the most intact ancient building in the city, with its open oculus — then Piazza Navona, the Trevi Fountain, the Spanish Steps.
Go to the Trevi early. By ten it is a scrum, and at seven in the morning you can be almost alone with it.
Afternoon: Villa Borghese gardens, and the Galleria Borghese if you booked it — timed entry, small, and the best sculpture collection in Rome. Or simply the gardens and the terrace at the Pincio over Piazza del Popolo.
Practicalities
Walk. The metro barely serves the historic centre, because tunnelling in Rome keeps hitting archaeology. The centre is compact and walking is how you find the things not on the plan.
Drink from the nasoni — the small cast-iron street fountains running constantly across the city. The water is clean, cold and free, and in a Roman summer that matters.
Eat away from the sights. The restaurants immediately around the Pantheon and the Trevi charge for the address. Two streets back, or across in Trastevere or Testaccio, is a different city.
If you have a fourth day
Ostia Antica — the port city of ancient Rome, better preserved than the Forum in places, and with a fraction of the visitors. Half an hour on the train and one of the most underrated sites in Italy.