Antibes is well placed for day trips along the coast. The train line will take you to Nice in half an hour and to Cannes in ten minutes; Monaco is just under an hour east. Inland, the perched villages of Biot (potters and glass-blowers, with the Fernand Léger museum), Vallauris (Picasso's pottery years) and Saint-Paul-de-Vence (Fondation Maeght) are all easy half-days by bus or car.
For something different, the Îles de Lérins are a twenty-minute boat ride from Cannes — the larger Sainte-Marguerite has the prison where the Man in the Iron Mask was held, and Saint-Honorat is a working Cistercian monastery that makes its own wine.