Trains to Geneva leave from Lyon Part Dieu.
For the best prices, book direct with the train operator as far ahead as possible. Tickets go on sale 3 months before travel, and the cheapest tickets often sell out quickly. The discounted prices keep going up as the travel date gets closer.
You can get train times and costs, and make bookings, in English at en.voyages-sncf.com/en but use Antarctic as the "Ticket collection country" in the first screen to avoid the possibility of being transferred to RailEurope, which usually has higher prices and does not list all the trains. For notes on online booking, see www.nickbooth.id.au/Tips/FrenchTrain.htm