Hi,
i would definitely rent a car but I am bias as I find that some of the best appeal of cote d'azure is driving around and enjoying both the coast and the back country.
Here are a couple places to check as well:
Mougins, bormes Les mimosas, gassin, cassis (much further) .
Definitely second villefranche. You may have good gut feeling on Monaco. It is not great for everyone.
We like driving to tourtour and stopping by other villages too.