10 Days in France - best itinerary?

Are you counting days or nights? For example, two nights typically amounts to one day and perhaps a fraction of another day. In terms of the itinerary, its a personal choice. Have you watched France Rick Steves episodes? Another option is to watch some YouTube videos, something like top ten Avignon/Provence. My reaction, for me, you need two full days to explore Normanday. Possibly less time in Avignon. Three full days in Paris, would likely be enough for folks less interested in Paris, although, we liked Paris much more than we thought we would.

A car would be quite helpful for both Normandy and Provence, unless you are booking day tours.

In 10 days you can see a lot. If you go with a car, you have more freedom to do as you please, also once you leave paris, the roads are not as crowded.
It might be a shame not to visit Paris, if you have not been. Even in 2 days, you can see quite a alot. After that Normandy is good and then down to avignon.

Thank you. And I should have said "nights." All three seem like "must-see" destinations, but I worry about fitting it all in at the expense of missing a remarkable experience.

I think three locations is doable. But I would pick three in a more compact triangle. Like Paris, Burgundy, Provence. Or Paris, Normandy, Loire. Just for context Rome to Rio says train from Bayeux (in Normandy) to Avignon (in Provence) is more than 7 hours.

How many people? With more than five people you would need two cars or a large van, which can be difficult sometimes to navigate through narrow streets in towns in France. Yet I prefer a car as giving more flexibility, both in Normandy and Provence. And I concur with counting nights on the ground, not days. If you are including arrival day as day one, be aware that that day is partially lost to arrival details and perhaps jet lag. I would not drive to Normandy, but take a train there (on day one) and rent a car on day two, after a night's sleep. Day 2 and 3 give to Normandy. A drive to Avignon from, say, Bayeux will take over 9 hours even on the autoroute, so day 4 is taken up by the drive. Even by train it will take at least 6 hours (train to Gare St Lazare in Paris, change stations, then Avignon from Paris Gare de Lyon), so day 4 is still a full travel day. Avignon could be 3 days (days 5,6,7). Return car at end of day 7, sleep in Avignon and take the TGV to Paris ( fast 2-2 1/2 hrs), so Paris is most of day 8 and all of day 9). Day 10 return home, or if you go home on day 11, then day 10 in Paris also.
In sum, your trip involves one day of mostly moving from one place to another, and another half day from Avignon to Paris. If your trip is truly 10 full days in France (arrival and departure days not included in your 10 days), then you have more wiggle room. In calculating drive times, be aware that most sites tell you a time that doesn't allow for stops for gas/food/comfort, or often more lovely rural roads. If Viamichelin.com says 4 1/2 hours to drive, add at least one hour to that. If it says 8 hours, add two hours to that. And factor in time to check out of lodgings, find new lodgings on arrival, find parking, and check in.

As for having a car in Normandy, I feel it is not as necessary as in Provence, but that means you will probably have the expense of a DDay beaches tour, which are not cheap. (Train to Bayeux, for example, guided tours once there, train back to Paris with transfer to station to go to Avignon, pick up car rental for Avignon leg).

With ten nights in France:
Nights 1,2,3...Normandy (Bayeux)
Nights 4,5,6,7...Avignon
Nights8,9,10...Paris
Hope this helps to put a perspective on your plans. It sounds doable as you have planned it, but I hate to see Paris shortchanged.

Best of luck!
Judy