They have champagne day tours and tastings out of Reims. But in Bordeaux you should get a car-- loads and loads of wineries in the countryside. You just see a sign for "cave" and drive in.
If you have time get over to the Dordogne area and see the castles and the caves with the 15000 year old cave paintings! (make a reservation). Not very far from the Bordeaux wine region.
About the only way to get to the Riviera from Bordeaux is by car, and it's a pretty good haul. I've said before on this board that I'm not a big fan of Cannes, St Trop, nor Monaco-- why did you pick these places? (I do like Nice) There is so much to do and see on the Riviera, and in much less ostentatious places. My favorite towns are Beaulieu sur Mer and Menton. Real places. Real people. Dip into Italy (just beyond Menton for a different culture, language and ambience). I can expand on my Riviera favorites if you'd like.