The south is a pretty broad definition. Along the Cote d'Azur, Nice is your most central option for day trips via train and bus. A car isn't needed (in fact, it can be a liability in some towns, unless you are heading to the countryside. It is tougher to do western Provence solely by public transport. You can base in a smaller place like St. Remy (or Nimes, Arles, the Luberon, Orange) and have then day trip by car to any number of places. Aix is a possibility but there is a bit more traffic around the ring road there and is a bit further south and east than many attractions. Smaller cities, towns and villages are often easier to get around, if you have a car. If you are thinking further west, Montpellier has some good train/bus connections (and is pretty easy to drive in/around) but further west, places like Narbonne/Carcassonne are best visited by car, if you want to daytrip.