I'm not sure about the mistral (wind) situation in Provence in March. I don't think it's as issue as far east as the Riviera.
The options for side trips by rail (along the coast) or into the hills (mostly by bus) from Nice are very numerous. You can easily keep yourself occupied for more than a week. If you stay in Nice you'll be close enough to hop across the border to Italy for a day if you want to. Among the popular places in that area are Menton (warmest spot, known for gardens but also has an atmospheric historic center), Cap Martin, Eze, Monaco, Cap Ferrat (Villa Ephrussi de Rothschild and Villa Kerylos), Villefranche-sur-Mer, Antibes, Cannes and St-Paul-de-Vence. I enjoyed the much less touristy hill town of Vence and the tourist-train trip to Entrevaux and Digne-les-Bains. There are many other possibilities.
Do be aware than March is by no means high season in that area. Many sights may have somewhat abbreviated hours of operation at that time. I recall seeing one or two very small, less-popular museums (not in Nice) that were open only 3 or 4 days a week. For indoor sightseeing it will be important to reseach opening days and times in advance.
The train along the coast is cheap, but it sometimes harbors pickpockets (though perhaps not in March when the pickings are a lot slimmer). If someone gets into your personal space unnecessarily, assume they are going to try to steal from you.
Rick has a guidebook dedicated to southern France. I urge you to buy it.