For a first time trip, it’s understandable that with so much to see and do in France, there’s a temptation to see and do a lot. France is as big a Texas, though, and that’s a lot of ground to cover in 9 days. If you’ve got 3 main destinations, though, and I’m assuming that Paris is maybe also part of the picture, if for no other reason than arriving and/or departing at/from Paris, you can do short visits to those 3 spots.
If you haven’t already gotten a Rick Steves’ France guidebook, that’s a great thing to have. If you land in Paris, see the city and/or rent a car or take the train or bus to Bayeaux, home of The Tapestry, as well as a Normandy Invasion museum and a nice cathedral. Visit Normandy beach(es) from there.
Head to Amboise in the Loire valley (drive? Fly? Train?), then to The Riviera in Nice, then back to Paris for the rest of your 9 days. If possible, fly “open jaw,” flying in to Paris, then out from Nice, to avoid the extra time returning to Paris.