One/two month stay location ideas.

I would like to suggest Lucca to you!
I have visited there and I thought it was a charming, wall-guarded town.

I also took a course at the local university about traveling to Italy and one of the people in the class asked the same question. Without us talking about it prior, the teacher also suggested Lucca.

I do recommend Airbnb and specifically this host in Lucca, she should have other listings. https://www.airbnb.com/rooms/1791364?checkin=11%2F01%2F2016&checkout=11%2F30%2F2016&guests=2&s=SlB10ljK

Take a look at SabbaticalHomes.com for Italy. This is a source of furnished rentals for longer terms.

I would love to spend a month in Venice and travel around the Veneto by train and bus, with the occasional car rental. Ideally, I would do this in mid-to-late Sept or October. Summer is just too hot and crowded. (We've spent about a month in Venice over the course of 8 trips, but I hanker for a long stay.)

I like the idea of Puglia, too, but public trans is harder there. I also think Sicily might hold some possibilities although I have not spent enough time there to recommend anything specific.

Maybe you could consider two locations, a month in each.

Walkability is pretty much a given in any place in Europe, but small villages, services and public transportation don't always go well together. Also not sure what you intend 'near water'. Lake? Sea? River?
I don't know if I'd spend an entire month or two in one place. I'd spend some time in the Veneto region. I presume you've been to Venice, so maybe a town on Lake Garda would be a good choice (Peschiera maybe? which is well served by trains, buses and ferries?). For Tuscany I'd recommend having a car and then stay wherever you wish. Same with Puglia (Apulia). Sorrento is a good place for Naples area. Frankly I wouldn't stay more than 1 to 2 weeks max in the same place.

I would second the recommendation for Lucca. Very flat city and everything is easily within walking distance. I would recommend staying inside the walls for the best experience. Train station is across the street from the "walls". Great shopping, churches,restaurants. Let me know if you need more information.

We spent two weeks in an apartment in Montepulciano with parking and loved it. The town is beautiful and has lots of nice restaurants so we would journey forth to various spots in the Val d'Orcia and wider Tuscany during the day and then had this beautiful town and its amenities in the evening. We'd do it again. Here is the countryside in September:
https://janettravels.wordpress.com/2011/09/20/patterns/
Usually you can get an apartment for the cost of 3 weeks if you book for a month. And in some rural spots it is even cheaper than that. Tuscany repays every day you can spend there and I think your idea of a smaller town is excellent. Rural gites are nice but you cannot drink and drive in Italy and it is nice to be in a town with restaurants, bars and such within walking distance. We once had a place in the countryside near Lucignano for a week that was about half a mile walk into the little town; that also worked. Alas have lost track of the link; we rented through acquaintances who owned it.

Ah I sort of skipped over the not wanting hillside hikes. We are in our 70s and were fine with Montepulciano but it is quite vertical -- everything is up or down a steep hill. The Lucca suggestion above is good for that as it is not a hill town but quite flat.

Spello is a beautiful Umbrian town and has a train station. However, it is on a steep hill.
For being near or on the water with train service, Santa Margherita Ligure( not a village) or Camogli.

I would split my time between Viareggio on the Tuscan coast and Peschiera on Lake Garda (as suggested by Roberto).

Trains in Puglia are ok. Just need to know many are not italo or trenitalia.
I would need to check the ones I used to travel there last autumn.
Umbria / Gubbio would also be very nice.

If you like seaside, try the Etruscan coast, Elba And Tuscan archipelago.
If you have a boat license, rent one and visit the little islands, taking a small picnic with you or eating fish on the beach.

I would recommend a car for such a long stay, particularly since you want to stay in a small (ish) town. Sicily could entertain you for two months easily, or two years for that matter. If you go on the internet, you should be able to find rentals everywhere, especially on airbnb. Many of their rentals are apartments, something I would want for such a long vacation. You can rent places close to Palermo or Siracusa in Sicily, both of which are large towns, but there are small towns close to them that are lovely and on the water. Palermo and Siracusa are fascinating places with a lot to see and do. September and October are excellent months to be in Sicily as it is still warm. On the mainland of Italy, the smallish town of Maratea and its neighbor Aqua Fredda are beautiful. They are on the west coast of Basilicata : don't confuse Maratea with Matera, which is more inland.