There are direct trains from GVA to Lausanne, the train station is a 5 mins walk from departures (to your left) follow the signs. You can buy traintickets from machines both in the bagage area or in the central hall of the station. Make sure you pay the full price (half price is only for kids or people with the half fare card), there's no reduction for seniors. If you buy through the app or online, beware that the reduced "supersaver" price tickets (dégriffé in french) are for the specific time and train schedule that you selected and thus not valid on earlier or later connections. The CFF app is very good for all public transport connections is CH (buses, boats, cable cars, metros you name it).
Fun fact: the toilets on the airport are free, so go to the loo just before you enter the station part