Trains from London to Chamonix

If you prefer not to fly, it's possible to get the Snow Train to Chamonix from Paris in winter - details from the Rail Europe site
Although this is a long journey, it's very well scheduled and runs overnight, so you can get the train from Waterloo after work, change in the evening in Paris and then go by sleeper.

Trains from Geneva Airport to Chamonix

Chamonix is unfortunately not easy to get to from Geneva Airport by rail - car hire, bus or transfer are much easier. However, the rail journey is very beautiful and well worth it if you enjoy train travel. There are two possibilities:

Direct from Geneva Airport to Martigny (3 hrs) and then take the Mont-Blanc Express which stops in Les Praz. This takes longer (and is more expensive) but has only one change.

Or take the train from Geneva Airport to Geneva main station, and then a tram to Eaux-Vives station in Geneva, where you can get a train to St Gervais, and then transfer to the Mont-Blanc Express which stops in Chamonix and Les Praz.

This is not quite a bad as it sounds, as it's well organised and the railway staff are very helpful, but not ideal with heavy bags. It takes about 2hrs 30mins to 3hrs 30mins and costs very little - about 8 euros one way. The Swiss railways site is much the easiest to use for times etc, .
www.sbb.ch

Trains within the Chamonix Valley

See Getting around Chamonix for more information.