Nice, France is on the mainline railway, allowing rail easy travel long distances (Paris, Lyon, Avignon, Marseille, Monaco and Italy). The train is also the easiest way to travel along the coast in the summer, for day trips to Frejus, Cannes, Antibes, Villefranche, Beaulieu, Eze, Moncaco, Roquebrune-Cap Martin and Menton.
Bus travel is available for some of the inland towns, like Grasse and Vence, but a car is necessary for the beautiful little perched villages: Peillon, Gorbio, Sainte-Agnes. Other day trips could include a tour of of St Paul, Vence, Tourrettes-sur-Loup, Bar-sur-Loup and the impressively perched Gourdon.
A single day's car trip will take you to perched medieval villages in the Tinée Valley (La Tour, Clans, Marie, Ilonse, Rimplas and Saint-Sauveur), but you'll really need several day-trips to spend at least a couple of hours in each of the villages. Past Saint-Sauveur, you could loop around to the west past Roure, Roubion and Beuil, and then back down the Gorges de Dalius.