As I'm sure you've seen in your research, any one of these cities could fill your entire trip on its own. But being your first trip, I totally get your itinerary.
I will make one recommendation. Unless you have a specific reason for spending three nights near Termini in Rome, reconsider this booking. The part of Rome you want to see (and stay in) can be thought of as about a 1 mile (2 km) diameter circle centered on the Pantheon. This includes the Vatican, the Coliseum and Forum, Trevi Fountain, Piazza Navona, Campo di Fiore, Trastevere, the Spanish Steps. Even Termini is at the very edge of this circle, but the area around the station is pretty bleak, whereas the area around Pantheon is excellent and the best place to first experience this amazing city. The closer you stay to the Pantheon, the more conveniently you're touring and exploring in Rome.
Anyway give it a thought and Happy Travels.