How much time do you have? And can you be more specific about what you mean by "spring shoulder season"? Spring weather north of the Alps as well as in the Alps is rather unpredictable, so I would definitely start in the south--flying into Nice or Marseilles--and work my way north. Even on the Riviera you are not guaranteed a great deal of sun in early May, though it's very inlikely to be cold.
One tip: If you land in Nice on May 1 (a major holiday), there will be no public transportation available into town with the possible exception of a few taxis. If you are interested only in Provence, not the Riviera, Marseilles would be a better landing point, but I have no idea about airport transportation there on the holiday. I sort of doubt that car-rental agencies are open on that day, either, but I have not checked.
If you opt to drive through Switzerland, don't forget that you will probably need to get a highway vignette for your rental car. I think it wouldn't be easy to cross that border twice and still avoid all highways.
It's going to be difficult to answer your question about time traveling between areas without knowing exactly which places you plan to visit in those areas. I have no experience driving in France, but viamichelin.com is probably the best source of driving-time estimates, though it is still thought to be optimistic. If you zoom way in on the Michelin maps, you'll see that some roads are highlighted in green. Those are the ones considered most scenic.
There are hundreds (if not thousands) of interesting places in eastern France. The car will allow you to move around faster, but I spent 89 days in eastern France last year and did not have time to include Paris. You might want to look for the book on the most beautiful villages in France.
