My first time to Italy, we went to Rome, Florence and Venice! I think it was a great introduction to Italy, and I think many people take this trip.
We flew into Rome, took the train to Florence and spent five nights there. (I had to be a bridesmaid in a wedding so I gave myself a couple extra days to see Florence). The big sights: the Duomo, the David, the Uffizi museum, etc.
Then we went from Florence to Venice, another short train ride and spent the weekend there (would have loved to spend more time there).
The only issue is the long train ride back to Rome, around 4 hours, but it is a nice break especially if you choose a quiet carriage. 5 nights in Rome, with one or two big things to see each day, i.e. the Coliseum, the Vatican museum, Spanish Steps, Trevi, etc. And the rest of the time meandering through Rome, eating gelato, enjoying the ambiance.
In total it was 12 days, it could be shortened, but there are so many big things to see in Rome that five days gave us time to see a lot of it without too much hustling.
Have a fun time planning!! :)