Avignon is a good home base for exploring parts of Provence. Just 2 km. from Avignon is Villeneuve les Avignon. There you will find Chartreuse du Val de Benediction, a 14th century cloister. The self-guided tour takes you into the hallways of this Carthusian chartehouse, past monks's cells, inner courtyards, and gardens. The most interesting thing to me was seeing the austere quarters in which the monks lived.
Pont du Gard is also just a short drive away. It is a Roman aquaduct, several stories high, built in 19 BC, which joined Uzes with Nimes. This portion of it is remarkably complete. You can walk up a steep path (about 1/2 mile) which takes you to the top, and you can walk across a lower level. It is hard to imagine how the ancient Romans engineered and built this massive structure. In the summer, there is kayaking in the river below, and kayaks can be rented.
Aramon, also within a few miles of Avignon is another walled city, though no where near as large as Avignon. It gives more of a feeling for the way the peasants lived, in the shadow of the Palace of the Popes. Arles and Orange are also easy day-trips from Avignon.