One full day in Nice. What would you do?

I would walk and walk around the town and along the beach. See different neighborhoods.
Take a short bus ride to the east to Villefranche for lunch or a train to the west to Antibes..
Try local Nicoise cuisine.
The Chagall Museum is unique as the space for each painting was part of the planning. His Russian Jewish heritage is prominent in his art.

Begin the day at the Chagall Museum then walk along the Promenade des Anglais. Afterwards explore the Vieux Nice (old town) and end the day at a wonderful restaurant where you can enjoy the southern French cuisine.

Walk around town toward and along the water and the promenade; visit Antibes in one direction and Monaco in the other; large art/antique market; climb up Castle Hill; eat some socca from a vendor, have a drink at the Hotel Negresco.

I like Mary Pat's suggestion, unless you don't like Chagall's work. It's a small museum and doesn't take that long to see. If the museum doesn't interest you, a half day in Vieux Nice and along the Promenade Anglais and a half day in Villefranche sur Mer would be good.

OMG lucky you, Nice is amazing. You might be surprised at how beautiful the Matisse museum is. It's spectacular. The setting is so beautiful, sunbleached red stucco and gorgeous views from the top of the hill. I visited Chagall then Matisse, then wandered to a beautiful cafe nearby and had a half carafe of cold white wine, a few nibbles, and read my Rick Steves guidebook and was so incredibly grateful to be there. Yep, I had ventured out on my own (the others stayed behind to do something else, can't remember what :) Just research the bus lines or easier yet, grab a cab or an Uber. Enjoy, Nice is truly one of my best travel memories and easily 10 years ago now....

Take in the morning flower market, wander through the old town and then go to the top of castle hill. Take lunch (reserve) at Le Plongeoir or Le Galet (or Blue Beach or Beau Rivage). You can take Tram 2 to the Port and then walk up Mont Boron, continue on to Fort Alban and either hike down to VilleFranche-sur-Mer or take the bus. You can dine there or take the train back to Nice. You can reserve for dinner at Bistrot d'Antoine, Le Sejour Cafe or Olive et Artichaut.

You can take the bus to Eze (le village) and take in the views and walk down the hill (study shoes) and take the train to Menton or Monaco. Or, head to Cap Ferrat and see the Villa Ephrussi. You can also take the walk around the Cap.

Thank-you everyone for your great suggestions. This site is so wonderful and I have used many of your recommendations for our trip to Provence. I am sure we won't be able to do everything and will certainly be back.