Montepulciano from Florence

Bus would be easiest. Check Rome2Rio site but prices may be out of date.

Train from Florence to Chiusi-Chianciano.
Then from the Chiusi station you have two options:
1. Bus from Chiusi to Montepulciano.
2. Train from Chiusi to Montepulciano station (on the Chiusi-Siena line). The Montepulciano station is down the hill at a distance from the actual town. I don't know if from there there are buses or taxis that can take you up to the town.

By bus you would change at Siena (train station).

The Chiusi-Montepulciano bus runs about hourly on weekdays, whereas the local train is less frequent and leaves you 5 miles away from the hilltown. www.sienamobilita.it should have bus schedules. Buses in Italy are usually reduced or non-existent on Sundays, but trains are schedules don't change much. There are also 2 buses per day direct from Florence to Montepulciano, see www.lfi.it. All info from p. 725 of Rick's Italy book.

IMO the train/bus combination via Chiusi is the better option, as there are several daily connection options (roughly hourly) compared to only a few if going by bus via Siena. There's generally less service on both routes on Sundays, but some Chiusi-Montepulciano buses do run.