We packed as much as possible into our four day trip. Protip: Look up the RomaPass. Buy it.
Night 1: We arrived at 9pm in Ciampino airport, and caught a shuttle bus to Termini Station. From there our accomodation was only an 8 minute walk away in Monti. We stayed at Residenza dell'Olmata, which I highly recommend. After dropping off our belongings, we went out for dinner. There were dozens of well reviewed restaurants only a few minutes walk away.
Day 1: On the first real day, we took our time but explored the city as much as possible. First up was the beautiful Basicilia Papale di Santa Maria Maggiore, which was literally steps away from where we were staying. Then we headed off to visit the Trevi Fountain (incredible), the Chiesa di Santa Maria nearby, the Sant'Ignazio Church, the Pantheon, Piazza Navona, the Basilica di Sant' Andrea delle Fratte and the Spanish Steps, stopping for coffee and food wherever looked nice along the way, and returning to Monti for dinner. Note: most of these sights are very close together.
Day 2: We visited the Colosseum, Palatine Hill and the Roman Forum. Afterwards, we went to the Jewish Ghetto by exiting at the western end (near the arch of Septimus Severus) up the stairs to Piazza del Campidoglio and from there down the Cordonata. At this point you simply cross the street (Via del Teatro di Marchello) and you're on the edge of the ghetto. The total walk is between a half and three quarters of a mile.
Day 3: We traveled to Ostia Antica, which is an ancient archeological site somewhat similar to Pompeii, but without the bodies. It was a ~40 minute trip by train. They've excavated the ancient city and there's some amazingly preserved buildings and sites. It's a lot of walking and we were there for like four hours, but it was super cool to see.
Day 4: On the last day, we visited St. Peter's Square, the Vatican Museums and the Sistine Chapel. Highly recommend paying for tickets to the Vatican in advance, as it looked like it would have been a nightmare to get them there. After this we headed to Trastevere for a look around and something to eat.
It looks packed, but for a regular guy and girl in our mid-20's it was actually fairly comfortable to fit all of this in. We were walking at a pretty relaxed pace, and obviously there's a lot of cool things to see on the way in between each of the major sights that I'm not even listing. We mostly relaxed in the evenings of Day 2 and 3. I even had time to revisit the Trevi Fountain a few times on my own on Day 3 and 4.