Montepulciano from Rome

The first step toward answering that type of question is a visit to Rome2Rio.com. It will show which modes of transportation are available between two points. You absolutely cannot trust the fares, travel times or frequencies that it displays, but if you keep clicking through the website you'll eventually reach links to the appropriate bus and trains companies. Following those links should allow you to find accurate schedules and--at least for trains--fares.

Be very careful about days of the week. Bus schedules in particular vary a lot, and sometimes there's no service at all on Sundays and holidays.

In this case Rome2Rio shows that there is no rail service to Montepulciano itself. You can take a train to Chiusi-Chianciano Terme and switch to a bus, or you can take a bus the whole way. Although Rome2Rio indicates that the travel time is similar, that may very well not be the case when you dig into the details. In fact, if Rome2Rio is correct that the first bus you'd use for the all-bus routing only runs once a day, that's not a mode of travel I'd want to depend on.

The train to Chiusi-Chianciano Terme takes about 1 hr. 40 min. The buses appear to be relatively frequent on weekdays, and the running time is 50 to 70 minutes. The question is what sort of alignment you'll find between the train's arrival time at C-CT and the bus's departure time from C-CT.

Here's the website for the Tiemme bus: http://www.tiemmespa.it/
There doesn't seem to be an English-language option, but if you use Chrome as your browser it will probably translate for you.

Here's the trenitalia website for the trains: http://www.trenitalia.com/tcom-en
Use Roma (Tutte le Stazioni) as your origin.

If by chance you haven't already seen Orvieto, I'd head there instead. The train gets there faster (in about 1-1/4 hours), and you don't have to coordinate with a bus. You just hop on the funicular that goes up to the historic district. I think it runs every 10 or 15 minutes. I believe there's also a local bus that goes up the hill.

I have done this. Your best option is train to Chiusi and bus to Montepulciano. Buses do run on Sunday, although on a reduced schedule. If possible, avoid arriving or departing on a Sunday so that you don't have to work your schedule around the buses.

I was just recently on the bus from Chiusi-Chianciano Terme to Montepulciano. You catch the bus next to the train station and purchase your ticket in the cafe/shop in the train station. When I was riding the bus back to CC after my stay in Montepulciano, the authorities came onto the bus halfway through the trip and checked to ensure everyone had a validated bus ticket, so keep your ticket handy.

Also, when you arrive at the bus station at Montepulciano, purchase the 1.20Euro “arancione autobus” - orange bus ticket at the coffee counter, which will save your feet, taking you up to the top of the very steep hill, and you can begin at Piazza Grande. It also stops at a few places as it climbs the hill in case you are staying at a hotel halfway up the hill.