No doubt you lose some flexibility without a car, but I do think you should be able to see plenty of countryside in train or bus. Just stick with the larger towns.
And I am sure that there must be some sort of bus day tour.
Just some ideas:

If you do Provence, maybe you could do Draguignon>Aix-en-Provence>Arles>St Remy de Provence>Avignon. It will be too early for lavender, but the gardens will still be glorious.

If you go to Burgundy, maybe you would take a route that includes Lyon?
Maybe Lyon>Beaune>Dijon>Paris? I know that Lyon is very walkable, and have heard that Dijon, is, as well. You could even dip into the eastern end of the Loire Valley on your way to Paris.

Look into “chemin de fer de Provence.” This is a privately owned (discount with railpass), narrow gauge rail line that serves many smaller places in Provence and Alpes Maritime. Here’s an example: http://www.beyond.fr…

Also, there is good bus service to even the smallest towns. You can take the bus from most of the stops on the Provence line to towns that have regular (sncf, the French national rail company) service.

Oldlady, is that the little yellow train?
Provence Beyond, which Oldlady linked to, is a very helpful website for France.

What is the little yellow train?
I’m sorry to have so little time on this trip! I thought three weeks would be fine! I’m finding that so much time is eaten up in travel from place to place. I guess I should not try to do Italy and France, I should just pick one, but I really must see my friend in France and Italy screams my name.

I will look into that website tonight and look around at other sites and see what I can come up with. I keep thinking I might have to drop the idea of stopping between Paris and Draguignan go straight to my friend in Draguignan. But I think Draguignan is not a very easy place to base daytrips from if my friend ends up being too busy while I’m there. I’m sure I’ll figure something out. Thanks for the suggestions!

Chemins de fer de Provence is sometimes referred to as CFP, but I’ve never heard a reference to Little yellow train. It uses a different station in Nice, referred to as Nice CFP as opposed to Nice Ville. There’s a good write-up about it in Jay Brunhouse’s Traveling Europes Trains. Here’s the Wikipedia article: http://en.wikipedia…. and here’s the official website. From the pictures, I doubt “little yellow train” applies… http://www.trainprov… At least some pages are in English if you click on the British flag.

In Burgundy, Beaune is a very nice place to stay! Old centre, and you can do wine tastings in cellars.