Travelling in Nice

If I understand your post correctly, you will have one week in Nice. Now given the time to do beaches, Old Nice, the museums, plus daytrips by public transportation to Cannes, Antibes, Monoco, Villefranche...you for sure won't have more than 2 days in which to use the car. That isn't enough time to drive into Provence, which is a long drive away from Nice in high summer traffic. So I would recommend that you take one day and drive to Vence and St. Paul de Vence in the hills above the coast, and a second day to drive to Eze--a spectacular hill town--and then around Cap Ferrat, which is not well served by public transit. You could do all of this if you had to without a car, but if you are determined to rent one for 2 days, these would be my suggestions. Hopefully other posters will weigh in with good ideas.

You certainly don't need a car for Eze-le-Village. It is an absolute piece of cake on the white bus. You certainly don't need a car for Cap Ferrat. There's a regular (smallish) bus which goes half way down the Cap, and its an easy really pleasant walk around the point, the walk is in the Book, and a very nice place it is. Parking, speaking from experience, can be tight in the off season - bet its a real killer in July - the two-way roads are really narrow and the one-way roads are even more narrow. S P de Vence is nice, and it is more remote but there's a bus going there too.

I am definitely someone who rents a car most of the time, but given what you want to do it will probably just be stressful. Now... if you want to spend only a day or two in Nice (which would be my choice) then I would rent a car after you have seen Nice and head for Provence and all the wonderful villages and beach towns in that area. But just to visit Eze or St. Paul de Vence (both are wonderful) I would use public transportation.