My suggestion is to stay in Nice. Try to find a hotel along the Promenade des Anglais. Renting a car is workable if you know how to park parallel in a really really tight space.
Otherwise, the bus is very convenient and economical to venture out to Monte Carlo, Eze, St. Paul de Vence, Grasse, St. Tropez, Cannes... etc.

Make sure to visit Villefranche-Sur-Mer, my favourite corner of the world! (only about 10 minutes by bus and 30 minutes on foot from Nice)

Feel free to email me if you need restaurant recommendation.

belle