Hi there. I live not far from Lausanne and used to take the train to Geneva in back a lot since I moved here. The SBB app is the best source to be informed as they update the schedule real time. You'll see even the delays if any which is not happening too often and not more than a few minutes.
1. The moment you land you can check the SBB app and have a clear idea what trains depart for Lausanne.
2. There are tons of trains from Geneva that pass through Lausanne as these 2 cities are a sort of railway hub for getting people from the Western part of Switzerland to the rest of the country. So it doesn't matter the direction as long it stops in Lausanne. Knowing the end stop will help before boarding so that you won't end up in France for example.
3. The Geneva Zurich train, for example, will arrive faster in Lausanne because it's like an express train with fewer stops. Brig train is a regional one, if I remember right, and have more stops between Geneva and Lausanne. However, it won't be a huge difference in terms of time so I would just pick the first train I can get to after passing through customs.
Extra piece of advice: be aware of the departure time and give yourself a few minutes in advance to get to the platform as Swiss trains and buses are right on time. 30 seconds later and you'll miss it. The train cars have the class written and sometimes they have both 1st and 2nd class on the same wagon. The IC trains, like Geneva-Zurich, have special places for the big pieces of luggage at the ends of the wagons. If not, you'll have to squize it between the row of seats especially if it is rush hour for the commuters. But I don't think it will be the case for you based on your late arrival on Geneva airport.