The train line does not go to Montepulciano.
You can train to Chiusi and hire a driver to meet you there.

Good wine tours are expensive. You can console yourself about the price tag if the day includes lunch, but to hire a knowledgeable guide and be able to sample several high quality wines -- plus the cost of gas! -- adds up, since the people who operate these tours need to make a profit.

If the prices of these tours don't make you blink, maybe one would work for you. You could ask to be picked up at the train station in Cortona or possibly Chiusi.

http://www.winetourscortona.com/en/index.php

http://www.winetouradventure.com/montepulcianotours.htm

Otherwise, if you take the train to Chiusi and get a taxi outside the station, for about 40e you can take a fairly scenic taxi ride to Montepuliciano. There are cellars in the center of Montepuliciano that you can visit and taste the famed Nobile di Montepulicano wine.

You'd need another taxi back to the train station (another 40e).

The train is easy, fast (about 1 hour) and cheap (7€-15€ depending on type of train) to either Orvieto or Chiusi. Wine tours operate from both Orvieto and Montepulciano. It's easy to take the funicular up to the town from the train station in Orvieto and meet your tour. In Orvieto, there is a wine school where you can taste that is sponsored by the consortium. They seem to only operate the tastings M-F.
http://www.ilpalazzodelgusto.it/en/regional_enoteca.html

For Montepulciano, you would have to bus or taxi from Chiusi. You could also book a local guide to pick you up at the Chiusi station and take you to a vineyard or two, Montepulciano and then drop you back at the Chiusi station to catch the train (make sure you know the time for last train to Rome so you don't miss it). I imagine a tour from Chiusi or Montepulciano would be much cheaper than a tour from Rome.

Here's one guides' website from the Montepulciano tourism page if you want to e-mail for some prices.
http://www.lamiasiena.it/eng/

Here's a list of the other guides from the tourism website with e-mail contact.
http://www.prolocomontepulciano.it/site-map-5

This one lists some prices for booking a guided tour. This agency books the housing for the language school in Montepulciano.
http://www.terretoscaneagency.it/catalogo/guide-en.aspx

These may be a bit more expensive (it's the wine consortium for Vino Nobile), but offer a variety.
http://www.stradavinonobile.it/booking/html/servizi.php?cat=1

It's also easy to taste wine and visit a cellar or two in the town of Montepulciano. Just wander in one along the main street in town.