Villefranche-sur-Mer makes a much better base than Eze-le-Village. There is frequent train and bus service through Villefranche. There is no train service to Eze-le-Village (not to be confused with Eze-Bord-de-Mer) and limited bus service.
Grier mentioned Cap Ferrat and the Plage de Passable. Here's a quote from a Lonely Planet guidebook: "Some 14km of marked walking paths cross the cape. A coastal path links Plage de Passable--a fine shingle beach (part public, part paying)--on the cape's western shore with the cafe-lined port on its eastern side....Other trails take you inland. Routes are clearly marked on a map distributed at the tourist office. On a clear day scale the 164 steps of the lighthouse on the cape's southern tip for a stunning panorama of the coast from Italy to the Esterel."
You can take the train to Cap d'Ail and then walk the secluded Sentier du Littoral (coastal path) from there to Monaco.
Antibes has a sandy beach, a walled old town, and an outdoor market.
You could take the Train des Pignes (Pinecone Train) from Nice up into the mountains to Digne-les-Bains and back. See here.
You said you want to take it easy when you're on the Cote d'Azur; but if you're art lovers you should know that, outside of Paris, Nice and the surrounding area have the best art museums in France.
