Avignon is a great starting point for walks in the lovely countryside of Provence. The city is at the center of an extensive local bus network (the bus station is minutes away from your flat).
I love walking and strongly suggest you take walks in nearby Alpilles ( Saint Remy with its Roman ruins and les Baux de Provence are only 25 km away) or even closer in the " Montagnette", stopping at the summit at the St Michel de Frigoulet's welcoming Abbey. Olive orchards ( photo), mediteranean pine forests, river and canals banks are on the menu.
Even closer is Avignon's huge and green Barthelasse island in the middle of the Rhone river, within easy reach by a free 5' ferry ride. There is even an olympic size swimming pool at the ferry arrival !