Whether or not you need a car is dependent on what you want to see. From Avignon, it's easy to get to Arles, Orange, Nimes, Marseilles, Aix, and the Pont Du Garde without a car. There's a few other places, but only a few; if you want to see small towns and rural areas, a car is best.
Rick's Provence And The French Riviera book has all the details about public transit in this area. His general France book has MUCH less info about the region.
From Avignon to Antibes there are several train routes. Use http://en.voyages-sncf.com/en/ to find them. I put in "Avignon TGV" as my departure station, but it pulled up routes starting at Avignon Centre as well. Avivnon Centre will be more convenient, as it's right in town, but these trains will require a change. You have to take a bus to Avignon TGV, but there are direct trains from there to Antibes. Since either route involves TGV's, you will get a discount for booking as soon as the tickets are released, As a bonus for doing that, you should be able to snag first class for little more than the price of second class.
Note that once in Antibes a car is not necessary, unless you really want to go off the beaten path. Trains run twice an hour between Cannes and Ventimiglia (with Nice in the middle, and Antibes between Cannes and Nice). Buses are much slower but are cheaper, and stop in places trains don't; they run three to four times an hour from Cannes to Nice, where you change to buses for towns east of Nice. Again, Rick has all the details.