From Marseille Airport to Aix there are direct buses running approximately every half hour. You will need to buy the bus ticket from a kiosk by the bus stops, not on the bus itself. The bus calls first at the out-of-town Aix TGV railway station and then at the Aix bus station, which is a little way from the town centre - you'll need to either walk or change to a local bus.
For Avignon to Marseille Airport you will need to take a train to Vitrolles station near the airport, from where shuttle buses connect to the train.
From Aix to Arles you can travel by train changing at Marseilles. It may be more convenient, however, to get the departmental bus route 29. However, the journeys from Aix to Arles only leave in the afternoon - at 1400, 1635, 1720 and 1815.
From Arles to Avignon you can get a direct train.