If you want to do countryside, wineries, open spaces, not rushing, then I would opt for a car because you are going to lose too much time getting on and off the train, getting the bus, etc. In Champagne and Northern Burgundy, terrain is pretty flat. Along the Rhone River and Avignon (Chateneuf du Pape), also reasonably flat. Can't speak to Bourdeaux region as I've not been there, but that's sort of out of the way from where you are going.

You will have to drive over the foothills of the Alps between, say, Nice and Genova, but the road is very good--not terribly steep or windy as long as you stay on the highway and don't venture onto the corniche.

As I recall, it's pretty steep around Genova but then flattens out as you get south of there along the littoral to Cinque Tere. Cinque Terre--the roads into some of the villages are somewhat steepy and windy, but not mountainous--just tricky to get down fairly close to the villages. If you plan to hike in the area, you could leave the car somewhere and hike in. If it were I, I'd skip 5T and turn north toward Alessandria right after Savona and hit the Po Valley.

As I recall, between Genova and Milano it's pretty hilly early on, but not terrible, and once you get to Piacenza/Pavia, you're in the Po Valley where the roads are flat, somewhat straight and very fast til you get to Venice. Also some wineries en route.

I envy you the time and ability to take this trip. Buon Viaggio!

The French countryside can best be visited by car. You need to choose which French countryside appeals to you: Burgundy, Provence, Jura? Guidebooks might help in this respect. There are also trip reports that would cover these areas; for mine, click on my name to find them.