Helo Lorie,
My advice is train, train, for heaven's sake take the train. France is Europe's largest country do not drive across it if you don't need to or some burning desire to see something you cannot get to on public transportation. Even then I would say take the train as close as you can and then rent a car just to go to that place.
In my opinion if you drive you will spend too much of your vacation time driving rather than seeing sights. Paris to Lyon is at least 5 hours by car and 2 peaceful hours by train.
Paris and Nice are both real cities so it sounds like you want to see some smaller towns but you don't give much indication of your interests beyond that.
-Lyon is fashionable, interesting city that is the gastronomic center of France with good public transportation and small core from a tourist standpoint. If food is your thing this is your city. It's not quite halfway to Nice timewise but is pretty close.
-Dijon would also work here - no personal advice but similar in size to Nimes but more northern influence.
If you want smaller towns the Provence region makes perfect sense - small charming towns on your way to the coast. You could do a northern city - Lyon, Dijon - train to Avignon and then continue down to Nice. You could also train all the way to the southern cities - Avignon or Nimes - see Provence and then continue on to Nice.
In the south you are more likely to get Roman ruins - both Nimes and Arles have stadiums and other sites - and bolder wines with cotes du Rhone style wines coming from this region. (The actual town of Châteauneuf-du-Pape is on the train line through this region.)
Avignon is also a good central hub so you could even train to Avignon and either stay two nights seeing Avignon and then the surrounding area - Arles, Nimes, or something like that. It really depends on your travel style and interests.
I personally prefer Arles to Avignon as a smaller provencial city but it complicates train transfers a little bit since it's off the TGV line and if you're looking for maximum time seeing places outside the travelling Avignon is easier.
Sample times:
Paris to Lyon - 2 hours; Lyon to Avignon - 1 hour; Avignon to Nice ~3:30hours;
Paris to Dijon - 1.5 hours; Dijon to Avignon ~3 hours; Avignon to Nice ~3:30hours;
Paris to Avignon - 2:30 - 3:00 hours; Avignon to Nice ~3:30hours;
Check the official train site or I prefer Loco2.com for tickets but there a many different train ticket sites,
And yes, I have driven in France on multiple occasions - including in Nice - but with limited time I can't recommend it.
Have a great trip!
=Tod