Coming to Courmayeur from inside Italy - it is possible to catch a train from central Turin and Milan to Pré-Saint-Didier - the closest train stop to Courmayeur. You may have to change trains in Chivasso or Aosta before you get to Pré-Saint-Didier. The name of the train company is: Trenitalia. From Pré-Saint-Didier catch a bus to Courmayeur. The bus company that runs this service between Pré-Saint-Didier and Courmayeur is: SAVDA.
Travelling by train is recommended as far as Aosta afterwhich a bus connection will take you to Courmayeur.
Coming to Courmayeur From France - travel to Chamonix Mont Blanc Train Station, serviced by France's national train system SNCF, and then get a bus to Courmayeur (SAVDA).