If you are doing the Villa Ephrussi you might consider the walk around Cap Ferrat (depending on the weather and time of year, of course). If you have a car, you can go to the towns of Touretttes-sur-Loup and Gourdon. You can try taking the Ferry to St. Honorat and seeing the Monastery and winery there. You can travel to Biot, for glass making and the Leger Musee. Go to Haut de Cagnes by train/bus and see the castle and Renoir museum. Go to Menton to visit the gardens and the Cocteau Museum. Travel up into the border mountain villages via the Train de Merveilles (Tende, Saorge) or drive to Peille and Peillon. Travel to Ventimiglia for the Friday market... or down to Borghidera for a taste of an Italian resort. Maybe Aquadolce for an Italian hill town. These are still pretty mainstream, but off the main tourist priority lists (such as Villefrance, Eze, St. Paul, Cannes, Monaco, etc)
Near your hotel is the Cafe Sejour. In the old town, try Bar des Oiseux. For a taste of the local scene, head to Place Garibaldi and along Rue Bonaparte (around Place du Pin - aka le Petit Marais) and into the Port for the new restaurants, bars and cafes.
ETA... just saw acraven's post... some duplication....