The car will be very useful for your stay in Montepulciano.
I would rent a car in Florence
You don't mention if you will be at the airport or train station in Florence but you can get a car from either location.
Airport driving out is a little easier as there are no ZTL fears and no busy traffic but I have rented from the train station before and is a straight shot out of the city walls so as long as you follow the directions you also avoid any ZTL concerns.
The drive to Montepulciano is fairly straight forward, if you want to make it scenic you can route yourself after Siena to Asciano to see the Crete Senesi countryside and from there to the Pienza area to see Val D'Orcia countryside before reaching Montepulciano. You don't need to stop in either town just providing town names so you can create a route map.
You cannot leave your car in Montepulciano and don't think trains go to Montepulciano anyway?
Look up a town on the way to Rome that does have a train station where that would be possible.
Rome Airport would be an easy drive / drop off but not sure you want to arrive at the airport in Rome?
Orvieto would be a good option but think only 1 company has an office there so check to see with the rental company you choose for Florence what your options are.
Chiusi is mentioned in the post above, and that is even closer to Montepulciano than Orvieto so that may good option for your dropoff.
Ideally I would not want to drive into Rome itself ; for the airport you do not have to.