Flights in/out of different cities

for Bayeaux, you can easily train from Paris or drive. We decided to train next summer. For Avignon, you can drive or train (TGV). I would pack only in a smallish suticase and a totebag/backpack, so that you can train easily. You can fly out of Nice, but going back to the US, you will need to connect. We usually depart out of a different city/country to not have to back track. If you drive to Avignon, it looks like Dijon is a nice place to stop. There are lots of small towns in Provence, so having a car would be a benefit. July 15 and weekends after that will be a bad drive to S. France, from what I read.
As it is your frst trip, I would stick mainly to trains. You might reserve a car for a day or two in Bayeaux and also in Avignon, but use the train to get there. Driving in a foreign country with French signs can be scary. Smaller back road are not nearly as intimidating. DH is renting a car next summer, as long as we stick to smaller roads.

Why fly? Take the train they are much more fun.

If you are travelling trans-Atlantic, you can fly into one city and out of another for just about the same cost as a simple round trip. Use the multi-destination search function. I suggest flying into Marseille's useful airport; the itinerary will require changing planes at a gateway airport, probably Charles de Gaulle or Schiphol. Marseille is worth considering itself or you can make an easy train commute from the airport to Avignon. Work your way north by train to Paris and fly home from there. You could follow the reverse order but you would have to catch a very early-morning flight from Marseille to connect to the trans-Atlantic flight home.
Resources: http://www.marseille-airport.com/ (especially its section on ground transportation.)
http://www.avignon-tourisme.com/home-1-2.html
http://en.voyages-sncf.com/en/
For budget European airfares, http://www.skyscanner.com

You may look into going to Avignon first upon arrival in June because once the Avignon Festival begins in July, it's much harder to find housing, restaurants, etc.

A month in France in summer - how lucky are you !!
Do you plan to hire or lease a car?
Perhaps break it down in to 4 sections of a week each.
After a week in Paris, the decision for the other 3 weeks could be tough ...
1) Alsace, Burgundy and Lyon.
2) Normandy and Loire Valley.
3) Dordogne and Bordeaux.
4) Provence and Cote d'Azur.

The most hassle free logistics if you plan on getting a car may be a connection on arrival in Paris (or elsewhere) to Lyon/Marseille/Nice. Return the car in your last big centre outside Paris, getting a train to Paris, and departing from Paris after your week (??) there.

Depending on costs, you might want to do a round trip to Paris. You can catch the TGV at the station at CDG and have a relaxing ride to Marseilles for about 40-45€ for a first class seat if you purchase tickets ahead of time. The station in Marseilles is in the heart of the city. Travel time is about 3 hours 45 minutes and I found it to be a pleasant journey. Current timetables show direct trains at 8:31, 9:58 and 11:58.

It's an ideal way to go if you might be stuck with a one or two hour layover at the airport. If you do fly, it's about a half hour trip to centre ville by public transport in Marseilles.

If you fly with Air France, you can include a train ride to certain stations as part of the flight ticket. This way, you are "protected" if your arriving flight is late - they put you on the next train. If you buy separate train tickets and your arriving flight is late, you have to buy new tickets - at last minute prices.

Here's the website explaining the program and listing the eligible stations (including Avignon): http://www.airfrance.us/US/en/common/resainfovol/avion_train/reservation_avion_train_tgvair_airfrance.htm

If you search for flights on the Air France website, it should show both air and train connections. Then compare these with other airlines' routes. Marseille airport gets service from lots of hubs - you could connect in Frankfurt, Madrid, Amsterdam, etc, depending on prices, flight times, preferred airlines and hubs, and which works best from your home city.

We started our Southern France adventure last year in Marseille. Very easy airport to maneuver through. Short walk to pick up a rental car, and easy access to freeways going in any direction you want to go. I would go with the suggestion above to start in Marseilles and drop off the rental car at the last stop before Paris.