Basic principles of trip planning:
1) Count nights, not days.
2) The day of departure is taken up with getting to the airport.
3) The day of arrival is often a jet-lagged haze, so don't plan anything important on this day.
4) Two nights in a place equals one full day. Three nights in a place equals two full days. And one night in a place is less than a full day.
5) Don't focus on what you can't fit in. Accept, right now, that something you really want to see will not work out on this trip, for one reason or another.
6) While it's great to get others' recommendations, be sure your trip is what YOU want to see and do. Don't go to a place just because "everyone" goes there, or some book or website says it's a "must see" (or even worse, "must see before you die"). With only 10 days, if a place doesn't make it to your personal A-list, it's out.
Specifics for Italy:
1) It's often easiest to see Italy in a line, north to south or vice versa (you've got the part covered).
2) Italy is insanely dense with history and worthwhile places, sights, artwork, food, etc. In ten days, you can only see a sliver, of a fraction, of a portion, of the country. Accept this now. Don't try to add more than can be reasonably seen and enjoyed in your very short time. And be careful of well-meaning people who say, for instance, "if you're going to Florence, you simply have to see Bologna too, because it's so close." "if you're going in Bologna, don't miss Ferrara." "If you're going to Ferrara, be sure to see Ravenna too." Pretty soon, you will have a great two-month trip - except you only have 10 days. Yes, all these places are great, but you have to ruthless in putting places on the "next trip" list.
3) A car is handy in Italy for small towns and countryside. It is an absolute liability for cities. So, before you plan on renting a car, make sure your itinerary is best served by driving. Or, conversely, if you're committed to driving, make sure you're seeing places that are easy with a car. For instance, if you want to go from Rome to Florence to Milan and stay in those cities, don't rent a car, or you'll be sorry (really - search the forum for posts of problems for the unwary driver in Italy). If, on the other hand, you want to go from Rome to small towns in Tuscany and Lombardy, only going to Milan to catch your flight home, then a rental car between Rome and Milan makes sense. If you want to stay in an agriturismo (farmstay), then a car is a near-necessity.
And most important: Assume there will be a next trip! Otherwise, you'll drive yourself crazy with what you're not seeing this time.