The main TGV lines (fast train) from Paris :
- Paris - Annecy.
- Paris - Lyon.
- Paris - Bellegarde.
- Paris - Chambéry - Aix Les Bains.
These stations have onward connections to Saint Gervais Les Bains Le Fayet (which is serviced by TGV during some periods of the year).
Tip: If you have trouble on the SNCF site finding a train, it may be because there are too many train changes. Instead of using "Chamonix" as your destination, try using "St. Gervais Les Bains Le Fayet." From wherever you come in France, you will almost always have to change trains at St. Gervais Les Bains Le Fayet (which is about 15 minutes down the valley from Chamonix after Les Houches).
If you are under 25 you can purchase a Carte de 12-25. This card costs about 60EUR and will give you up to 50% discount on train travel - great value. The SNCF also offers a Carte Senior with similar discounts if you are over 60. More Info (in French).