The most efficient transportation to southeastern France from downtown Paris depends on where you want to go. There is no really fast TGV service down to Nice (the fastest train takes 5 hr. 40 min.), so flying isn't a crazy option at all. However, if you want to include in your itinerary a bit of Provence (Aix-en-Provence, Marseilles, Arles, Avignon, or Luberon villages), the train will be faster and a lot more comfortable. Avignon and Marseille are the fastest fo reach, but there are trains to both Arles and Aix taking under 4 hours. Reaching villages will require onward road transportation (see below). The train trip east from any of the large cities in Provence toward Nice will be somewhat slow, because the rail line in that direction doesn't allow TGV-speed service.
I haven't been to St. Tropez, Cannes or Monaco. If you want to observe glitz and boutiques by the sea, a visit to one of those places might be fun, I guess. St. Tropez is one I'd take a pass on unless I had a specific reason for choosing it, because it has no train station; you'd end up needing to include a boat or a bus in your travel plan. It would just be a lot easier to get to Cannes or Monaco.
I didn't like Marseille as much as I expected to. I chose a hotel in a not-great location (near the main rail station) and found the museums quite disappointing. Nonetheless, I enjoyed walking around on both sides of the old harbor.
A car is very helpful in Provence if you want to see the cute villages and small towns. Train service is very efficient between the larger places I mentioned above.
Along the Riviera coast, a car would be a pain. Traffic can back up severely and unpredictably, I was told by a local. The train isn't fast (and you need to be alert for pickpockets), but it is very convenient. Most of the Riviera-area hill villages and towns people want to see are accessible by bus ( a few by train), so whether it's worth it to rent a car depends on what specific places you want to see. Parking a car is likely to be challenging and expensive.