My parents live in Avignon so I get to ride around that area quite a bit. I’m afraid I don’t have any recommendations where you can hire a bike, but as for riding in the area, it’s fabulous. If you’re pushed for time, I’d stay on the road. The area around Eygalieres is very pretty – stick to the hills. But if you cross the Rhone, the really great stuff is in the Luberon. Cross over to Cavaillon and head over to Bonnieux from Taillades via a disused road right along the top of the Petit Luberon. Anywhere on Google maps which is green is really very nice.
If you’re going Ventoux way, I really enjoy riding down to Sault, then down the Nesques Gorges – fabulous day out. Distances are pretty great so you might want to make use of the local buses.