My friend and I are going to spend a week in Provence. Can we do it mainly on public transportation? We will rent a car for a day to do the hill towns of Luberon but do we need the car longer? Also, what is the best location for a central place to explore the regoing? We would like a smaller town with good connections as we are planning on the Roman sites, Carcassonne, Luberon and then onto the Basque region.
Whether public transit is sufficient depends on what you mean by "Provence." If you mean Marseille, Cassis, Aix-en-Provence, Arles, Avignon, Nimes, Orange, Pont du Gard, and a very few other places, then yes, trains and buses can work fine. I saw all of these except Cassis and Pont du Gard without a car; some did have some gaps in transit, but all were fine (I stayed 10 nights, and used Marseille, Arles, Avignon, and Aix as bases). However, if by Provence you mean the place with the little towns and the lavender fields, then you will need a car.
It depends, of course, but for a week I'd want a car. You'll see so much more, especially the countryside and smaller towns. The Luberon is going to be very difficult by bus. I want to note that Carcasonne is not in or really near Provence and you should plan to see it perhaps as an overnight enroute to the Payees Basques. Without a car, there is still great stuff to see, including Roman antiquities in Arles, Nimes, and Orange, all easy by rail, and the Pont du Gard, doable by bus. You'd probably want to stay in Arles or Nimes in that case. But with a car there are many charming smaller towns where you can stay. If you are there for a week a rental can be a great deal, or you could split your time between the Rhone Vally and the Luberon.