Hi, Stewart -
Lucky you, to be able to hop over to Nice for a weekend. From here it's a major journey.
I second Lucy's recommendation of the market. On the day when it isn't a flower & veg market, it's brocante...bric a brac...all of it great fun. There are good cafés for when you want to watch the passing parade instead of participating in it.
On food days, look for the people at the very end of the tables, towards the back of the market. That's where you find the home-made chèvre and other goodies.
Saturdays are popular wedding days in Nice, especially in spring or summer. The Hôtel de Ville is just across from one of the entries to the market - it's fun to watch the bridal parties coming and going (for the civil ceremony that precedes a church ceremony in France.)
Definitely use the train for inter-city trips, though there is also very good bus service up & down the coast, from a conveniently-located bus station. I like Villefranche-sur-Mer, a very short bus or train ride from Nice - it would make a good excursion.
Julie (In Seattle, dreaming of la Côte d'Azur)