Since you won't be able to go via Switzerland (unless you purchase a Global Pass) the easiest route is as often in France via Paris. It looks like a big detour on a map but all high-speed railways lead to Paris. You need to transfer from Gare de Lyon to Gare de l'Est so it's a bit annoying, have at least 1h margin for the connection. Change in Bellegarde (and sometimes Annemasse) afterwards.
If you want me to have a look let me your travel date. For example on winter weekends + holidays there are direct TGVs from Paris Gare de Lyon to Annemasse/Evian-les-Bains.
This is a good route which avoids Paris and Switzerland :
- TGV Strasbourg - Mâcon 09:05 - 12:18 10-20€
- TER Mâcon - Bourg-en-Bresse 13:04 - 13:34
- TGV Bourg-en-Bresse - Bellegarde 14:12 - 15:00 10-20€
- go drink a coffee at the station or something
- TER Bellegarde - Annemasse 16:15 - 16:55
- TER Annemasse - St-Gervais-les-Bains 17:02 - 18:16
- TER St-Gervais-les-Bains - Chamonix 18:28 - 19:13
A bit of an adventure on hidden lines, esp. Mâcon - Bourg-en-Bresse !