We are a retired couple wanting to rent a home or apartment in Italy for two months. We would like something close to the train to do day trips or two to three day adventures. We have been to many of the larger cities, Rome, Venice, Sorrento, Florence, Cinque Terra, Siena, Como, Bologna. We want to explore smaller quaint towns but will need easy access by train. We are looking for recommendations for a lovely central place to stay by restaurants, shops and transportation. Any suggestions?

My suggestion would be somewhere in the Po valley as the web of railway lines offer you lots of opportunity. You've been to Bologna which is a central hub perhaps have a look at Ferrara or for the very quaint (and I mean that very positively) why not Montagnana?

Check out Spoleto. See if this link to my TR works:

Turin, Spoleto, Bologna and Bellagio TR

I haven't been, yet, but Orvieto looks really interesting. Decent sized hilltop town on the train line between Florence and Rome.

What time of year? For two months, I would think beyond trains. "Smaller quaint towns" don't always have train service. But if it must be only trains... Here's a map of train lines:

https://www.italiarail.com/italy-rail-map

You'll see the NE has the most train service, and there are a few mid-to-small-sized towns that are rail hubs... Novara, Vercelli, Alessandria, Asti, Piacenza, Pavia. Of these, I've only stayed in Asti which we enjoyed a lot. They hold a palio similar to Siena's... first Sunday in September. We've passed through some of the others. No strong opinions on any of them...