Nice has the Chagall, Matisse, Fine Arts, and Modern and Contemporary Art Museums, the 4 mile Promenade des Anglais, Old Nice, Castle Hill, etc. I have never lived there but I know many one-way rides cost only 1 euro regardless of length for up to 74 minutes of travel in one direction, including transfers. That includes bus 100 from Nice to Monaco taking 45 minutes--about 4 buses an hour Monday through Saturday and 3 an hour on Sundays. Last return bus to Nice is about 20:00. Unless it has changed recently bus 100 stops in the same area of bus 81, 200, and 400 near the corner of Rue Alberti and Blvd Jean Jaures. There is lots to see in Monaco like the Prince's Palace, Changing of the Guard, Napoleon Collection, Cathedral of Monaco, and the Cousteau Aquarium.
Returning to Nice you could take bus 112 with 7 buses a day from Monaco directly to Eze-le-Village but there are no buses on Sundays or take bus 114 with 5 to 6 buses a day from Monaco to La Turbie then return to Nice by bus from either of those places. Villefranche-sur-Mer with its Citadel and scenic walkway and its old town is only about 20 minutes from Nice on bus 100 with 3 or 4 an hour but I would avoid the days that a cruise ship is in port there. I really don't know anything about Antibes or Cannes on the other side of Nice on the coast or about St Paul or Vence above Nice but those are all places close to Nice that you could research here on Trip Advisor. I not only like to read people's comments but I like to look at the photos they have posted on Trip Advisor of those places.
Edited: 24 April 2013, 04:33