We have a similar timeframe in Rome in September, we'll be staying in the Testaccio neighborhood, so our details include our metro rides and stops; we'll have my fit and active, yet 80 year old, mother with us, so while this is an ambitious plan, it also allows us to modify if needed should she get tired and not feel like walking as much.
Day 1
Walk to Pyramid of Caius Cestius, metro Piramide--Colosso, we'll start with the Roman Forum and Palatine Hill, then do the Colosseum, and from there walk to Circus Maximus, then return (walk) to apartment. Going in that order allows us to bail, if needed, after the Colosseum and simply return to the apartment via the metro at Colosso.
Day 2
Piramide-Termini-Cipro metro ride to Vatican Museum, Breakfast at the Vatican (7:15 AM entry) - Museums, Sistine Chapel, St. Peter's Bascilica; we anticipate we'll be there from 7:15 AM until about 1:00 PM. When we're done there, we'll take a break, grab a bite to eat (about an hour), then head over to Castel Sant'Angelo and the Supreme Court area, then make our way to the Mausoleum of Augustus and then Piazza Popolo, and then to the Flaminio metro station to return to Piramide station.
Day 3
We'll take Piramide to Spagna station and start the day at the Spanish Steps, walk over to the Trevi Fountain, then head over to the Palazzo Borghese until lunch. Take a break, grab a bite to eat. Then we'll walk over to Piazza Navona, then the Pantheon. After that we'll check out the Roman ruins at Largo di Torre and make our way over the bridge to Trastevere and see Basilica of Our Lady of Trastevere, walk the neighborhood, making our way to Porto Portese and the bridge back to the neighborhood we'll be staying.
Day 4 is our departure day from Rome, and we have a car, so we'll head out via the Appia Antica for a few hours, stopping along the way as we find interesting things to see, then head up to Parco dei Mostri in Bomarzo after lunch on our way to Florence.