Traveling in Tuscany for 2 weeks from home base in Bagni di Lucca.

Driving is not too hard in this area, but parking will add to your costs, and driving into Florence doesn't make sense. Think about doing most trips by train, but renting a car for just a few days for Greve or other hilltowns that are harder to reach by train and bus.

You'll have to decide whether two or three train connections one way is "too much for you, " or whether the schedule really works for a daytrip. For instance, Bagni di Lucca to Camucia-Cortona station takes 4 hours, 2 connections, and leaves at either 7:00 or 10:00 a.m. Rome also takes 4 hours each way. How to Look Up Train Schedules Online gives you the DB train schedule link and tips for using it. All schedules are published through June 14 and some later.

You are within very easy reach of Lucca, Pisa and the CT. All can be done by train. From Bagni di Lucca, you are about 2.5hrs from Florence by train. You definitely don't want to drive in Florence with their many, many ZTLs (restricted driving zones). A rental car is a great idea for Greve, Chianti, Siena, Voterra and Cortona. Just don't park inside any of the city walls. Again, the dreaded ZTLs.

Thank you, Laura and Larry for your suggestions. This brings me to another point. After Italy, I will be going to Provence, France. I will be staying in St Paul de Vance and will probably need a car there. Can I rent a car for the last week in Bagni di Lucca, and then drive it to Provence, where I will probably need it for 2 weeks?

If I were staying in a small place like Bagni di Lucca, I would rent a car.
The Bagni train station is actually not in Bagni (it's in Fornoli, about 5 km away) on the Aulla-Lucca line, but I don't know how close you are to it. If you aren't walking distance, you need to wait for a bus.
Also the train is ok for visiting Lucca or even Florence, since you won't need a car in either and both are doable by train. But Greve? Volterra? Cortona? even Siena? You can drive to those places in a couple of hours from Bagni, but with public transportation you'll need at least 5 hours each way and I couldn't even guarantee you could make it back on the same day.
Regarding your trip to France, you need to realize that when you rent a car in a country and return it in another, there are exorbitant drop off fees to pay, which range in the hundreds of euros. It might be a good idea to return the car within Italy somewhere, proceed by train to France, then rent another car in France.
If you plan to go to France then return back to Italy, then it's not a problem. However you need to notify the rental company you will be traveling outside of the country where you rent.

Just check on the rental car rates and get prices for a few days vs. a weekly rate. Often times it is cheaper to rent for the week. That way you don't have to cram all your driving into a few days. I usually rent at the airport in Florence and it is easy enough to get out to the freeway so you don't end up in the 'zones'. It is really nice to have a car when you are in the countryside. You may decide to take an evening drive just to watch the sunset over the vineyards! Plan for lots of roadside picnics:))

If you rent a car, check the rates from the airport or from a city location (for example on the consolidator autoeurope.com). Picking up a car at the airport usually carries higher rental rates.
If you land in Florence and go directly to Bagni, Europcar has a location on via Forlanini, which is very close to the airport, but it's considered city location, therefore it has lower rates. If you come from Rome, then it might make sense to visit Rome first, then rent a car from a Rome downtown location.

Thanks for the valuable advice.
It sounds like it would it be easier for me to rent a car for a week in Italy, drop it off in Florence or Pisa, and then take a train to France. Then I could rent a car in France.
Only problem is, I am confused about how to get to St Paul de Vence by train. Which ticket or rail pass do I choose? I keep on seeing France to Italy, but not the other way around. I know that it can't be that complicated, but I am afraid of getting stranded at some rail station in the middle of the night.

Also, should I arrange the car rentals in Canada before I leave? I have also read something about leasing. Is it really cheaper to do that for 2 weeks?

For your transportation from Bagni di Lucca to France, you can drive but I would highly suggest turning in the car at Ventimiglia to avoid huge drop off fees in another country. Then, simply hop on the French train at Ventimiglia, take the short ride to Nice Ville and rent another car there. Just walk up, buy the tickets, validate them and get on the next train. It's really that easy. Or, you can take the train from Bagni di Lucca to Nice Ville. This will take 8.0hrs-9.5hrs with 3-4 train changes along the way. If you drive, its a simple local train ride from Ventimiglia to Nice Ville.

Unfortunately there are no rental car locations for major rental car companies in Ventimiglia, near the border.
The closest is AVIS in San Remo. Otherwise both Hertz and Europcar have offices in both Genoa and Savona.
From any of those cities you can catch a train to Ventimiglia and from there one of the numerous regional express trains to Nice. In Nice you will then rent another car.
It's generally better to make arrangements from home. You can use a consolidator, such as AutoEurope.com which generally get better rates rather than directly from the major rental companies' websites.

Thank you so much for all of the advice. I am not feeling as nervous about my trip. I will check with the car companies for the best price. I will actually be in Bagni di Lucca for 2 weeks starting September 1st, but thought that I would travel by train/bus the first week and then rent a car for the second. Is that the easiest and least expensive way to go? The villa that I am staying at does not provide parking. Dropping the car off close to the French border sounds like an excellent idea. Now I will have to work on my itinerary!