You can't connect to Montepulciano by train. You must take a bus and it would probably be more convenient from Siena than from Florence. There is a Montepulciano Stazione with train stops, but that is not where you want to go as it's miles from Montepulciano and you can't count on any local transportation from there to Montepulciano.

Buses for Siena province which includes Montepulciano/Siena/Pienza, etc. (not Florence) are on Siena Mobilita. You will find more bus options from Siena to the wine towns of Montalcino (Brunellos) and Montepulciano (Vino Nobile). From Florence the bus options will be more of the Chianti area. On Sundays, bus service can be very limited for some areas.
http://www.sienamobilita.it/EN/index.html

On the train line between Florence and Rome, the easiest stop would be Orvieto. Perhaps arrive Rome and immediately train to Orvieto (it's only an hour by train from Rome). Overnight Orvieto and pick up a rental car. Drive through southern Tuscany (Montepulciano/Pienza/Montalcino), stay in one of the towns a few days and do day trips. Drop the car in Siena (visit? stay?) and then on to Florence a couple of days and end in Rome.

Great advice!