I'd not recommend getting from Geneva to Chamonix by train as not only does it take longer but it's the inconvenience of train changes, especially when you're lugging around snow sports gear and luggage.
First of all, you have to get yourself to the station Genève-Eaux-Vives then change at Roche-sur-Foron or Annemasse. The train from Geneva airport does not go to St Gervais/La Fayet from where it's necessary to join the line going to Chamonix Mont Blanc.
However, the travel challenges to get to ski France, Italy and Switzerland on the one pass makes it worth the effort. Click on the thumbs below to read more, but the fastest way to get from Geneva to Chamonix is via the Autoroute Blanche with pre-arranged shuttles.
There's now a direct train from London to Geneva which will reduce the hassle of train changes and takes just over 6 hours. But I'd investigate flights as there's any number of discount air services to Geneva Airport.
Nonetheless, at the Geneva end, you still need to have pre-booked transfers to Chamonix.
The TGV train terminates at St-Gervais-les-Bains-le-Fayet and you transfer onto another train system that services just the valley. You’ll find skiers commuting between villages from Servoz to the other end of the valley at Vallorcine. This train will take you directly to the main town.
The journey takes anywhere from 2.5 hours plus, depending on connection times. For a more direct selection of routes when traveling from Geneva to Chamonix by rail, insert instead the destination St Gervais-les-Bains.
The other route via Montreux and Martigny is really the long way
around, although the journey is spectacularly scenic.