Ciao Shrishti,
I understand that driving in a foreign country is never easy, particularly when the driving is on the opposite side of the road!
A cab or car with driver from La Spezia all the way across Tuscany would not be cheap!
Montepulciano is not easy to reach on public transport either, although there ARE buses from CHIUSI.
The easiest solution would be to break up your itinerary in Tuscany into steps, moving from CT into Tuscany with stops along they way. For example, from CT to Lucca or Pisa, then on to Florence, from Florence on to Montepulciano (train to Chiusi and from there bus to Montepulciano). That way it works better with your overall itinerary using solely public transportation.
How many days total do you have for Tuscany? When are you coming? What other places did you want to see? The entire Val d'Orcia is a gorgeous part of Tuscany, but like Chianti, the train does NOT cross it and the main towns, just works its way AROUND it. That is the main reason why we recommend car rental and driving to explore these two areas. Another option: use SIENA as your base and from there take a guided tour (with transport included) to Montepulciano and other parts of Val d'Orcia and into Chianti. That way NO DRIVING and you still get to visit both areas!

Take a look at some of the guided tours you can find:
http://www.partner.viator.com/en/8357/