What's the simplest and cheapest way to get from central Nice, France to Monaco by public transportation?
Bus 100 (Nice - Monaco - Menton) is a very good option and costs a mere 1 €. It runs every 15 minutes mostly, and the trip takes 30-45 minutes. It takes an absolutely scenic route along the coast, passing through seaside towns such as Villefranche and Beaulieu.
(There's also express bus 100X which takes the motorway and costs 4 €; obviously not as good for sightseeing.)
(NB: when I took this bus in Nov 2012, the place of departure had temporarily changed due to construction work: instead of Station JC Bermond, it left from in front of Musee d'Histoire Naturelle, 60 Boulevard Risso. Ask around if needed, and you will surely find it.)
Arriving in Monaco, there are 6 stops to choose from (do get off at one of these as the bus continues towards Menton):
Heading from Monaco towards Nice, the route is slightly different; then the Monaco stops are:
The buses don't run very late: last one from Nice leaves at 20:00, and from Monaco at 20:22. First departures in the morning are about 6:00. You'll find details and full list of stops in bus 100 timetable.
Alternatively, take the train ("Nice Ville" to "Monaco Monte Carlo"). It is faster (22 mins) and arguably easier, but costs a little more at 3.60 €. There are about 2 regional trains per hour (daytime), according to SNCF. A good thing is that the trains run much later; last one leaves from Nice 22:25 and from Monaco 23:12 (on weekdays).
The above applies to regional trains (TER) which would probably be your primary choice on this route, though you could also catch an intercity/TGV.