Italy is pretty much my second home. Definitely don't skip Rome. To outsiders, Rome might seem dirty and congested, but it's over 2,000 years old. It's a magical place and this time of year it won't be too overcrowded. Most tourists make the mistake of going in summer when it's just insane.
Florence is pretty wonderful too, but people forget the natural beauty of Italy. Don't try to do everything at once in one trip. There is just too much to see. 3 places for 10 days is great. You can go to Pompeii in a day from Rome, and that is well worth seeing. Hell, you can even do a tour group which is what I did last year, and it was fantastic.
If you are able, take a couple of days to see the Amalfi Coast. This time of year is gorgeous. Ravello, Positano, Maiori, Minori, etc. are gorgeous. Take a day trip over to Capri or Ischia by boat, or a drive over to Paestum (gorgeous Greek ruins from a colony that was there about 300 B.C. I think). Venice is nice too, but again, allow yourself a chance to get away from the really touristy places. Outside of Venice you can do a day trip into places like Balsano da Grappa in the Veneto region and see the Palladian villas. I did that this year and it was just awesome.
Another great day trip is Tivoli, outside of Rome. Check out Villa D'Este for sure, beautiful gardens.
Here are some recommendations for Rome:
Ice cream -- San Crispino (right by the Trevi Fountain) is the best, hands down. Be sure to try the honey combined with any of the sorbets. Everything they make is what is in season, Giolitti is also very good.
Restaurants (you can Google all of these to get locations)
La Matricianella -- my favorite place, you can't go wrong there. Wonderful pasta, and the green peppercorn steak is unreal.
Perilli -- In the Testaccio neighborhood. Great traditional Roman cuisine.
Taverna dei Mercanti -- in the Trastevere neighborhood. Wonderful rustic place (I believe it's 16th Century) that has fantastic steaks and pizza. Also try the polenta w/sausage.
Lots of very, very good restaurants in Trastevere and Testaccio. Also be sure to stop by Cafe Greco for coffee, which is near the Spanish Steps.
If you want a great location to stay in Rome that is inexpensive and quiet, check this place out:
http://www.hotelsuisserome.com/