I'd personally spend the whole 10 days in Paris. But having lived there, I might be biased!
I agree with the person above who said that Paris is best enjoyed slowly.
In terms of other cities, I think Bordeaux is pleasant for an overnight or two, but certainly not critical for a first trip to France. If you go, head out to Dune Pilat, which is very neat. Nice was also nice, but it takes a LONG time to get there via TGV from Paris (something like 5.5 hours, I think, since the Marseille-Nice section is not TGV speed). Nice has some good museums (Matisse, Chagall) and also has the great Foundation Maegt nearby. Still, I would probably pass considering you only have 10 days in France and will spend half a day getting there and half a day getting back.
Also, I don't remember the SNCF route network offhand, but is there a good way to get Nice-Bordeaux, or do you have to go back via Paris? I know there are trains that could probably do Nice-Bordeaux directly, but not sure if that is "true" TGV on that line.
If you really want to get out of Paris, maybe a trip to Aix (which is under 3 hours direct on TGV) or Strasbourg (I think 2.5 hours) would be good for 2 nights.