Audrey,
Nice would be a good intermediate point between Avignon and the Cinque Terre (assuming you're not stopping there when you first arrive). It's a beautiful city and there's lots to see and do in that area.
Having a car isn't an advantage in the Cinque Terre, as parking and vehicle access are limited. There's a large car park at one end of the "new town" in Monterosso, but I'm not sure what the costs are. You won't be able to drive into the town, so if staying in Monterosso your best bet would be to get a hotel at that end of the new town or the far end of the old town and use the local trains to get around. I can offer some hotel suggestions if you're interested. If you were planning to stay in Vernazza, I'm not sure if there's parking available at all. It was destroyed in the floods of 2011, and I don't believe the facilities have been re-built yet.
Note that for driving in Italy, each driver will require the compulsory International Driver's Permit, which is used in conjunction with your home D.L.
It would probably be a good idea to book accommodations in the C.T. as soon as possible, as they tend to book up very quickly.
Happy travels!