Did a variation of this a few years ago when I was over there for Euros. Basically spent the best part of a fortnight travelling around the South of France by train. It was absolutely fantastic.
Bordeaux is a really nice city - it’s got quite a relaxed and friendly atmosphere about the place. Actually liked it so much I went back the next year. Toulouse and Montpellier are worth a look, too. Carcassonne’s old town is really picturesque, like walking round a film set for Robin Hood, but it does get clogged with tourists.
The trains are pretty good (if they’re not on strike) and good value compared to the UK. SNCF has quite a good app which is useful for journey planning. Be prepared to get irked after a while by the little jingle they play before the platform announcements. The TGV is rightly admired, but I think I preferred the slightly slower Intercites which have old fashioned carriages with comfortable seats and big windows so you can enjoy the view.