I suggest to take the train FR1 (the cheaper one, 5.50 euros) to Roma Ostiense. From Ostiense you can have a walk to the quarters Aventino, Celio and to the central archaeological area (Coliseum and Forum). The same train stops also in Trastevere, and Trastevere would be another good option - but the historical quarter Trastevere itself is not so near the railway station Trastevere, you'd have to take the tram 8. It depends on what you prefer to see, the archaeological sites and the green oasis of Aventino or the popular soul of Trastevere (without forgetting some amazing monuments like the church Santa Maria in Trastevere). Anyway, I would avoid the train Leonardo Express, too expensive and not so useful for your trip.
I would suggest taking the train to Ostiense, then the metro to the Colosseum stop. Don't get tickets to enter any of the sites, but walk around and see the Colosseum, Constantine's arch, the Roman Forum (from the street above) and the Imperial Forums and Trajan's Market (on the other side of the street from the Roman Forum. Then take the metro on to Termini station, change lines, and get off at Barberini. From here you can walk to Trevi Fountain, then on to the Pantheon (definitely go in) and Piazza Navona. After this, walk to Corso Vittorio Emanuele II and take the number 64 bus to the Vatican. Admire St. Peter's Basilica and the Piazza. You should definitely try to go inside; it's free and the security line moves fast.
At this point, if it's not time to head back to the airport yet, you could walk to the Tiber, turn right, and walk along the river to Trastevere. You might even have time for a quick meal. Then take a taxi to Trastevere station and return to the airport. I would count 2 hours for ancient Rome, 1 1/2 hours for the historic center, and 1 1/2 hours for the Vatican. Add 2 hours for travel to and from the airport; it should take less, but just to be safe, pad it.
I would suggest taking the FM1 train (not the Leonardo Express) to the Trastevere stop. From there, take the number 8 tram to Largo Argentina (the last stop). The Pantheon is a short walk from Largo Argentina. There are only two train lines from the airport, so just be sure not to take the Leonardo Express. The tram stop is outside the station, across the piazza. Get the tram with destination Largo Argentina.
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