You could spend months in Rome and not see it all. If this is your first trip, I'd keep it to one week, or sensory overload takes over. So take those extra nights and add them to these cities.
3N Venice.
5-6N Florence.
Spend 2-3 in the city and do day trips,allowing you to spend your evenings in some of the most beautiful Piazzas in Italy.
Day trips by train:
Bologna (50min).
Montecatini Termini (46m) and Pistoia (11m). Combined as day trip.
Lucca (1H19m) with Pisa (31m) also combine well.
By bus: Siena 1H11m. Full day.
Greve in Chianti 1 hr.
San Gimignano a tour would be the easiest.
All of these towns have great squares with plenty of cafes and people watching.
6-7N Rome.
In Rome, these Piazzas are some of the most lively. Piazza Della Rotunda by the Pantheon, Piazza Navona, Campo di Fiore and in the Trastevere neighborhood, Piazza di Santa Maria. But there are Piazzas all over the city.
1N Ponza. The time it takes to get there warrants a nite, maybe two.
5N Sorrento. Do Pompeii, Amalfi Coast, Capri and maybe Ischia
2N. Naples see the Archeological Museum and Capella Sansevero and as many othe sites as you can fit in. Its a warren of squares and Piazzas that as you turn a corner, they're just there. Do try to spend some time in the city itself. While it can be chaotic and gritty it has its charms.
Fly from Naples to Greece. Airport is 6 miles from the center.
Edit: Adjust your days according to what really speaks to you.