Voting with the folks above that 9 nights = only 8 full sightseeing days plus one partial jet-lagged one: not nearly enough for 4 locations + the CT and hill towns. If you are a first-timer to Italy, 4 nights is my recommended minimum for Rome as there is so much ground to cover there. I'd give Florence 3 and spend 2 in either Lucca or Siena. Skip the CT as you'll only have two days for Florence (there is much to see in that one as well) and late winter weather could be a crapshoot in March anyway. Ferries won't be running, lots of businesses will be closed, and day-tripping that one puts you smack in the middle of the day-tripping mob if very large ships are in port in La Spezia.
Night 1 - land in Rome, train to Florence/partial, jet-lagged day
2 - Florence - full day sightseeing
3-Florence - full day sightseeing
4-Siena or Lucca - hotel transfer/partial day sightseeing
5-Siena or Lucca - full day sightseeing
6- Rome - hotel transfer/partial day sightseeing
7- Rome - full day sightseeing
8- Rome - full day sightseeing
9- Rome - full day sightseeing
If it were me? I'd choose just two bases, Florence and Rome, and do some easy day trips from Florence. That would cut down number of hotel moves + give you some flexibility to work around the weather. Chilly and wet? Stay in town and cruise the churches and museums (although the Uffizi really needs advance, time-entry tickets to avoid a long ticket queue) Sunny and dry? Hop a bus and go to Siena for the day or take a train to Lucca. You could do a 3rd day trip (escorted wine tour?) from Florence but gosh, that would leave you little time to explore that city's many treasures unless not all that interested in them.
Night 1 - land in Rome, train to Florence/jet-lagged partial day
2- Florence - full day sightseeing
3- Florence - full day sightseeing
4- Florence - full day sightseeing
5- Florence - full day sightseeing
6- Rome - hotel transfer/partial day sightseeing
7- Rome - full day sightseeing
8- Rome - full day sightseeing
9- Rome - full day sightseeing
Food and wine can be had anywhere so it really doesn't matter where you stay for that.