From Geneva airport to Lausanne station

To answer the easy question first: the Genève-Aéroport train station is attached to the airport. It is an extremely short walk, even if carrying luggage. In fact, walking is your only option.

Swiss train stations have ticket vending machines, and the main stations (including Genève-Aéroport) will have a staffed ticket counter as well. For the casual traveller, there is no point in buying an advance or electronic ticket.¹

Swiss trains don’t have seat reservations,² and the train will never be full to the point where you can’t find a single seat. The ticket you buy will be valid for any train on that route. The fastest option will be the IR (Inter-Regio), but technically you could use the same ticket for any number of connecting trains along that route. Schedules are available at http://cff.ch/, but the trains run frequently enough during the day that you need not worry about it.

¹ Technically, you could hop on any main line train without a ticket and pay on board, though you may be subjected to a surcharge. You cannot do that with buses and regional trains, which are clearly marked with an eye logo, indicating that the honour system is in effect, and you must already have a validated ticket.

² … except in special cases, such as French TGV running in Swiss territory, or overnight trains.

A tip for saving a few bucks on the train ticket:

On the baggage claim area there are machines that give you free tickets to Geneva. You just need to press a button and they’ll give you one. They are valid only for arriving passengers, so make sure to keep your boarding pass with you. Then, in the machine for buying the train ticket you can select “change departure station” and pick Genève instead of Genève-Aéroport. This way you’ll pay CHF 22.40 instead of 27.00.

There is a machine for buying train tickets in the baggage claim area, so you can do it while you wait for your luggage.

Once on the train, if the inspector asks for you ticket while on the Airport-Geneva section, you show them the free ticket together with your boarding pass. If you are on the Geneva-Lausanne section, you show them the ticket you bought.

You don’t need to buy the ticket online (there are automated ticket machines as well as manned ticket booths), but you’ll save some time, and possibly some money: even though the standard ticket is 27.- CHF, you can buy discounted tickets for as little as 16.20 CHF. Drawback: discounted tickets are valid only on a specific train, so you’ll need to time things carefully (too early and you’ll miss the train, too late and you’ll just waste time, so it really depends on your priorities).

Note that if you are going to stay for a little while in Switzerland and move around a fair amount, you might be interested in getting a Half-Fare Travelcard (“demi-tarif”). It costs 185.- CHF for a year, but get you a 50% discount on most rail travel in Switzerland (and more).

As for getting to the train station: once you exit the baggage claim area, turn left. Walk. Turn right. Oh, you’re in the train station.

Yes, the train station is at the airport as it’s been already said.
If you’re tight on budget, rideshares like blablacar offer rides for around 5 € one way.

I think it’s a great alternative to the way overpriced swiss train system.

There are four trains per hour from Genève-Aéroport (the train station is directly at the airport) to Lausanne. The travel time is about 45-50 minutes. There is no discount if you buy the ticket in advance and a seat reservation is not possible on these trains, so even if you can buy the ticket online in advance, you have no disadvantage if you choose to buy the ticket at the station just before departure. In any case, the ticket is valid on a chosen date and not limited to a specific departure.

If you don’t have any discounts, the ticket price is CHF 27.00. You also have the option to buy a ‘City-City-Ticket’ for CHF 34.20, which also gives you the right to use public transport in Geneva (zone 10) and Lausanne (zones 11 and 12) on the day of travel. Even if you don’t need public transport in Geneva, the City-City-Ticket is cheaper than buying a regular train ticket and a day card for Lausanne (CHF 8.60) if you are planning to use public transport there after arriving.