Hi Lisa!
On a recent trip to Italy we did in this order Venice- Florence- Cinque Terre- Amalfi Coast - Rome. All of that was over a solid 2 weeks. We had 2 days in Venice, 3 in Florence and 3 in Rome. I think 3 in Florence and 3 in Rome are the absolute minimum because there's a lot to do and see!
In Florence, we got the Firenze Card (worth the $!) and thus were able to cut all of the lines to do the duomo climb, see David in the Accademia and the Uffizi all in one day. Then, the next we climbed Piazzale Michelangelo and did the Boboli Gardens (they were amazing- a must!!). Since we didn't have much time rather than travelling into Tuscany and needing to drive, I opted to pre-plan a great guided tour for a day through nearby Chianti region. This allowed us to really get a sense of the area with no worries.
The company I used was "I Just Drive" who partnered with my hotel (the hotel was the Davanzati...it was my favorite stay in all of Italy- highly recommend! Their service was unreal and they even hosted a daily happy hour for all of their guests). The tour was called "Chianti Best." It is capped at 8 people max and on the day we went it was just us and another couple. They picked us up right at our hotel and we got to visit Siena, San Gimagno, Monteriggioni, and then ended at a lovely vineyard with tasting of wine, olive oil, and balsamic vinegar and snacks included.
In Rome, 3 days meant we had a walking tour I booked for the afternoon we arrived (focused on pantheon, spanish steps, etc) a day to see the Vatican in the morning and then a free afternoon after that, 1 day for colosseum/palantine hill, etc. By the way in Rome, I booked all of my tours through Walks of Italy. They were all fabulous! I especially recommend the "Pristine Sistine" tour because you get into the Vatican 30-40 minutes before the majority of the public, when it gets packed. Also loved their VIP colosseum tour because you get into the underground of the Colosseum and also the very upper tier that you can't otherwise see. To me, tours make a trip because it's all about the experiences and learning about what you are seeing.
So, in a week I'd do Rome and Florence (and book 1-2 tour day trips from Florence). Venice is also doable in 1-2 days...it would be ideal if you can be in Italy for 8 days to add that in!
Hope this helps!