We've stayed in AirBnB's and hotels in several places in Provence and one place in Dordogne over four different trips to France. (I love France and speak some French.) As others have said, you can't go wrong with either. It's probably easier to see all the main sites in the Dordogne than in Provence in one trip. We find that staying in one place, with day trips or the occasional overnight to see the points of interest works best. Moving every couple of days introduces too much hassle and time spent on logistics.
Which region you choose depends on what you most want to see, but you'll need a car for both:
Provence: wine tasting, Roman ruins, Pope's palace, market towns, Provencal cuisine
Dordogne: ancient cave paintings, castles along Dordogne River, gorgeous scenery, foie gras, wine tasting if you go to St. Emillion/Bordeaux.
If you like small towns, we enjoyed staying in Uzes, which is just west of Provence and is driveable to Roman ruins, Arles, and Isle sur le Sorgue (great market on Wed, Sat, though my step-daughter thought the market in Lourmarin was the best in Provence). Uzes has a lovely market and lots of good restaurants.
In the Dordogne, we stayed in the lovely medieval village of Limeuil, in a small, affordable inn called Au Bon Acceuil that has an excellent restaurant on site and helpful inn keepers. We would stay there again. We also liked the town where the cave paintings museum is located, Les Eyzies de Tayac (sp?). We found accommodations in Sarlat le Caneda to be above our budget, although Rick Steves recommends it.
To get to Provence, you can fly into Paris and take the TGV (bullet) train and in 2.5 hours be in Avignon, where you can rent a car; or you can fly into Marseille. To get to Dordogne, you might get to Bordeaux somehow and drive from there. (We arrived in Dordogne via car from Provence, but it's quite a drive and I don't recommend it unless you have a lot of time. We had three weeks.)
Have fun!