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Nice has a lot of busses taking you for 1 Euro to many destinations along the coast such as Villefranche, Beaulieu, Eze, Monaco, and Menton to the East, and Cagnes, Antibes, Juan les Pins, Cannes and others to the West. Trains are a little more expensive but faster and go to the same places.
If you speak some French, you could also participate in some of the guided walking tours in Nice (about 1 to 2 hours) covering themes like baroque or classic architecture, art deco style, empire style etc.