Central Location

Those places may be too far apart to use just one central location. I like Avignon as a base which would also reach Arles easily and provide a quick exit to Paris. I have stayed at Hotel du Parc a few times, although it has been 4 years since I was last there. The rates were very reasonable. The hotel is close to the central train station and bus stop. There are other reasonably priced hotels in the area. Be sure to reserve your train seats in advance of the date you want to use them. Some of the trains fill up.

Mary, in September we stayed at an excellent B & B in the southwest of Avignon. It's an easy walk through the city wall to Avignon Centre train station, and from there they have a shuttle to Avignon TGV, a very nice modern station.
Not sure if you are thinking in euros or usd for your budget, Maison Boussingault was 120 euro per night. It was very comfortable, excellent breakfast on the terrace in the mornings. Anne is an Avignon native, and was extremely helpful to us.
We went from here to Lyon for a few nights, then on the way to Paris we stopped in Beaune for the day, arriving in Paris about 8:30 that night.

http://www.tripadvisor.com/Hotel_Review-g187212-d2290093-Reviews-Maison_Boussingault-Avignon_Vaucluse_Provence.html

You need to stay overnight at most stops, but can choose either Avignon or Arles with a daytrip to see the other one. It would be possible to stop for a few hours in Carcassone in the middle of a long travel day from Avignon to Lourdes (leave at 6:30, 7:30, or 9:30; arrive before midnight), but you may prefer to spend a night there and leave in the morning (half-day visit). The direct train from Lourdes to Paris takes 6 hours with no backtracking toward Avignon.

How to Look Up Train Schedules and Routes Online gives you the DB train schedule link and tips for using it.

Rick's hotel suggestions are in his book, except that he doesn't cover Lourdes.

My daughter and I spent 3 nights in Avignon in a B&B through the BedyCasa.com lodging network - it was a homestay with Marie, not far from the train station, and it was really great. Cost in high season is 65 euros for 2, and breakfast is a bit more - or you have access to the kitchen with a fridge, in case you want to fix your own. Marie was an interesting hostess, and I'd be happy to stay there again.

If you are interested in homestays, I recommend BedyCasa; just be sure to check the reviews, as always, and be prepared for the fact that you are staying in people's homes, not a hotel. We stayed at one place in Montpellier that was fine as long as you like cats [which we do] and are fit... it was on the 5th floor with no elevator. Just a lot of stairs!