Welcome to the forum Taty!
Is that 10 NIGHTS? Usually to get 10 days you need to be IN Italy for 11 nights. Just checking because the number of nights matters.
Rome, Florence and Amalfi Coast are all top destinations and will be crowded- even at end of October. There is hardly a shoulder season anymore. San Gimignano and Siena are also the top 2 hill towns visited by all the tour groups, bus trips etc. They will both be crowded.
If you want to avoid crowds as much as you can then spending time in quiet parts of Tuscany would be my choice.
Pienza, Montepulciano- the Val D'Orcia region.
Assuming 10 nights- which equals 9 full days of sightseeing.
Land in Rome- take train to Florence - there is a direct fast train FCO to Firenze SM Novella at 1:53.
If that time doesn't work then Leonardo Express to Termini and catch the next fast train to Firenze SM Novella
Spend 3 nights in Florence. That gives you 2 full days for sightseeing- as long as one of the days is not a Monday you can see a lot. As you might remember Florence is very compact and walkable.
Pick up car as you leave Florence- lots of advice on that here- just ask. (All drivers must have an IDP)
Spend 3-4 nights somewhere in Tuscany- our favorites are Pienza and Montepulciano- just make sure your lodging has easy to access parking.
Then drive to Rome to drop off your car- can dor at either airport or in the city at Termini or Tiburtina station.
Or drop at Chiusi- (Avis/Budget) just. a short drive from Pienza or Montepulciano and take train into Rome
Rome 3-4 nights.
This way you don't have a split stay in Rome and it's one less hotel change.
You will need to prebook all the major venues in both Florence and Rome.