Nine nights goes by quickly. It may be wise to spend five in Paris and four in Nice and simply fly between the two. I flew that leg last September, it was 50 euro one way on Air France.
Lyon is one direction, Dijon another and Limoges still another direction. You will have to pick the one that appeals to you most. I liked Arles better than Avignon, but Avignon is more of a transportation hub. I haven't been to Marseille because friends told me it wasn't nice.
Train will be more expensive. It's possible to train direct from Paris to Lyon, from Lyon direct to Avignon and from Avignon to Nice. With your limited time, you really don't want a rental car because you won't have time for sights outside the immediate area of your stop.
I normally like to stay two nights to give me one full day for touring. If you leave Paris early, you can visit the Gallo-Roman ruins/Basilica in Lyon, walk the city center, get dinner in a Bouchon restaurant, then stay one night. Ideally you could leave early for Avignon, walk the center and head for Nice. It looks like the train to Nice is a longer ride (with a stop in Marseille). If you can't leave Avignon fairly early, it might be best to plan a night there.
If you are searching for trains using the SCNF sight, be sure to scroll through the day. Sometimes a direct train won't show in the time of day you search but runs at a different time. Direct trains can save a lot of time and hassle when you find the right one.