Monaco is an easy jaunt to the east, on the opposite side of Nice from Antibes. The day we visited it wound up pouring rain, so we spent almost all of our time in their great aquarium, started and original directed by Jacques Cousteau.
Just northwest of Monaco, at La Turbie, the Trophy of the Alps (Trophée d’Auguste), the massive classical remains of a once-bigger 2,000 year old monument to the Roman emperor Augustus’ military victories over combatants from the Alps. Quite the historical sight. There’s also a museum if you’d like a closer look.
North of Nice, and outside of Saint-Paul-de-Vence, is the Foundation Maeght, a wonderful Modern Art collection. If you went to the Picasso museum in Antibes, you’d like this. If you didn’t go to the Picasso museum before, consider another visit to Antibes.