I am considering taking my two older children (ages 18 and 22) for a week in France before my wife and two younger children (11 and 15)come for a week or so in Italy. I am looking for a combination of a nice place to relax and sort of fit into a local lifestyle for a time but also somewhere that we can take day trips and where my kids won't get bored.
A week in Provence is an attractive plan.
I'd want a car, and with a car I'd find a house or apartment in small town, not a city like Avignon.
I think another poster may have suggested St. Remy, which would be an excellent choice (but not the only one) for the kind of experience you describe. Great day trips to Arles, Avignon, Orange, Les Baux, Pont du Gard, Luberon towns, Aix, Marseilles, Camargue, Nimes, Sorge valley, etc. may tempt you away from small-town life.
When you are done you can return the car and take the train (or fly) to Italy.