Train vs Driver

Not a good question for this site. Most folks here tend to focus on budget or certainly value travel and will recommend trains or buses. Doubt if anyone here has done that but it is an interesting way to go if you can afford it. We had a car in that same general area. While the scenery is nice, pretty of rolling hills with grape vines and olive trees, it also very repetitive. It doesn't change a lot so using a car just for scenery would be a marginal reason. But the convenience factor of a car is very high.

Rather than focusing on a driver for the transits between the various cities you might be better off just taking the train to get from one place to another, settling in for a few days (as I think you plan to do anyway), and booking day trips from there to explore the countryside. Would certainly be cheaper than using any sort of private transfer service just to get from one point to another ... which the trains can do very well and at nominal cost.
Once in Montepulciano, for example, there are a host of day trip providers who can craft a pleasant day exploring the scenic back roads and hill towns of the Val D'orcia, assuming that you don't want to just rent a car in Chiusi and drive yourselves that is.

I agree with Robert. Use the train/bus combination to get from Rome to Montepulciano and on to Florence, and save your "driver" budget for day trips from Montepulciano. The scenery in that area is beautiful and there are several interesting nearby places that can be challenging to visit using public transit. I'm guessing you'll have at least 2 full days in Montepulciano, so you could do 2 half day trips or one full day trip and still have time to enjoy the town.

Edit: My reply assumes you don't want to rent a car, but renting one would be a good option for what you want to do.

Take train from Rome to Chiusi.
Rent a car in Chiusi.
Roam around Montepulciano and wherever you want to visit in Tuscany.
Return the car in Florence (or in Chiusi then take a train, whichever is more convenient for your plans.)

Roberto hit the nail on its head. Montepulciano is not serviced by trains (nominally they have a railway station, but is very distant from the town and on a local line with sparse service) and the closest station with intensive service is Chiusi/Chianciano Terme. While there are connecting bus, having a car is much more efficient. If your budget allows, it may be a reason to use the services of a driver. But remember, once you are in M. you either use buses, or drive, or you are almost a captive there.