A car is pretty essential in the Sth of France. There are car hire companies at both Avignon and Nîmes stations and at the airports.
If you arrive at Avignon TGV, just walk out the back of the station, and down the stairs to the rental agencies. All the major car rental companies are represented there. The most frequently used companies are: Avis www.avis.com, , Budget www.budget.com, Europe car www.europcar.com, Hertz www.hertz.com
It is much easier and cheaper to book online or over the phone before you arrive. Most agencies don’t like taking arrivals without a pre-booking too. You can use this link to help plan your drive to Uzes from Avignon: https://www.viamichelin.co.uk/viamichelin/gbr/dyn/controller/Itineraires?strStartCityCountry=1424&strStartAddress=&strStartMerged=Uzes&strDestCityCountry=1424&strDestAddress=&strDestMerged=Aix+en+Provence&choiceType=1&ie=UTF-8&x=38&y=14 or I can send you a pdf print out if that is easier.
Another great website for general information is the Frenchguide https://au.franceguide.com/home.html?NodeID=1&xtor=SEC-6