We did this in 2015...took the TGV from Paris to Avignon, picked up a car at the train stn, then drove to Carcassonne. Stayed two nights. We didn't arrive until later in the evening tho, as our train was delayed for a few hours, so we basically just went to eat, then back to our airbnb. We spent until early afternoon at the site, then a little time walking around the town. After Carcassonne, we went back to Provence, then headed on to Nice, where we dropped the car. Hindsight - one night would have been enough, and we could have taken off the next afternoon as the town itself didn't hold much interest for us.
We've never had a vignette in France, unless something has changed since 2015. You most likely will need an International Driving Permit (IDP). Driving was pretty easy. You can avoid toll roads, but it's going to take a lot longer. For example, when we were heading to Carcassonne, we had the GPS set to avoid tolls. After an hour of what seemed like endless round-abouts, we switched it to allow tolls to see how much faster it would be - it shaved at least an hour off the arrival time, so we went that way. Some tolls were expensive, some not so much. There are websites that will give you a good idea of the tolls (is it the Michelin site?). Another great site for figuring out travel times is rome2rio.com - they have an app as well and has been so helpful in our planning. If you aren't in a hurry, you can just take diff roads.
Some people would rather use public transport - and generally we do as well, but when trains are not as frequent, and buses are a pain in the arse - sometimes a car is just the way to go. It was great for us to be able to get to places that aren't as easy to get to using pub trans (Les Baux, Millau Viaduct), or to stop somewhere along the way if it catches your eye. For us, the car was a better fit for that part of France, more flexible.
I'd suggest doing what we did, getting the car, heading right to Carcassonne, then make your way back and drop the car back off in Avignon and get the train back up. And some people don't care for Carca - I wouldn't go back, but we did enjoy our day there.