You can take the train, but not sure about costs, and it's slow:

Cross Geneva to the Eaux-Vives rail station and take a French SNCF train to Annemasse or Roche-sur-Foron. Change here for Le Fayet/St. Gervais Les Bains. Change here onto the Mont Blanc Express and a panoramic ride to Chamonix. Journeys will take between 2 hours 51 minutes & 3 hours 25 minutes.  A shuttle transfer by road takes about 1hr 10 mins. 

There are lots of shuttles from the airport - try Cham Express or Alpy Bus. Between 25 and 40EUR each way.

There are also two buses a day from the Gare Routiere (Geneve centre, in Paquis) at 9am and 5pm going to Cham, not sure of times coming back. You can find all details on www.gare-routiere.ch – click on tickets and reservations, then alpine resorts.

hey, if you're looking for cheap travel, try www.blablacar.com and if people are already going, you can arrange a car share. I used it when i needed a ride back from bordeaux back to geneva. Good luck!