All trains between Nice and Italy run frequently and all stop at Monaco/Monte-Carlo.
The station is an easy 10-15 minute walk to the Grimaldi Forum.
It is possible to get the Eurostar from London to Paris then the TGV between Paris and Nice.
Trains between Milan and Genoa run reasonably frequently during the day.
Overnight trains to and from Strasbourg, Paris, Toulouse, Irun, Port-Bou, Milan, Venice, Pisa and Rome are also available.
Regional Express Trains connect the many towns along the Côte d’Azur.