You can visit Orvieto and Assisi using public transportion by taking a bus to Chiusi train station (about 30 minutes) and then a train to Orvieto (about 25 minutes) or Assisi (about 1:45). You can also take a bus to nearby Pienza, visit Siena using a combination of bus to Chiusi and train to Siena (about 1:15) or visit Cortona using a combination of bus to Chiusi, train to Camucia-Cortona (about 25 minutes) and bus up to Cortona (about 10 minutes). Most other places will be inefficient to visit from Montepulciano using public transportation. You might be able to take a taxi to and from your wine tasting and dinner if it's fairly close to town, but Uber is not a possibility. Renting a car would simplify your transportation and open up more possibilities, but it does come with the risk of traffic fines and you should never drive after drinking (very low limits).
Edit: Chiusi has some interesting Etruscan sights, so you might also consider visiting there. You would probably need to take a bus or short taxi ride up to the main part of town where the sights are located.