From Nice there are tons of possible daytrips by train, bus or car. A partial list includes Monaco, Menton, Eze Village, La Turbie, Villa Ephrussi, Villefranche sur Mer, Haut-des-Cagnes, Antibes, Juan-les-Pins, Cannes, Vence, and St. Paul de Vence.
None of them "require" more than a few hours, as they're small; if you are taken with a place, you will of course want to spend longer. The buses in the area are each €1 (yes, one euro, specifically to encourage people not to drive). The trains cost a bit more, but are faster; for longer trips such as Nice to Cannes, the bus seemed eternal, and I was very glad to take the train instead. The trains also run later.
For more information about this area, including places to go and detailed directions on how to get there from Nice, get a copy of Rick Steves Provence And The French Riviera. His general France book has much less information on this region.
For the places you specifically asked about: from Nice to Eze Village and Eze Village to Monaco, you will be taking buses, and will need to check a schedule (when I was there in 2008, there were no buses between Eze and Monaco on Sundays). Between Monaco and Nice, there are 4 buses an hour and two trains an hour, so that connection is easy.