I'm not a fan of Monaco, but it certainly gets a lot of hype. Of the 20+ museums in Nice, do not miss the Matisse and the Chagall. The coastal train that runs from St Raphael to the Italian Border is cheap and fast. From Nice-Thiers station you can go East or West and have a good time.

My favorites are East: Villefranche, Beaulieu, Menton, and do not fail to go to Ventimigilia, the last stop, which is in Italy. It's got a different look, different food, and a totally different culture even though it's just over the border, only 20 mins on the train from Nice.