Train question

Yes, you buy a ticket to the first station in Switzerland, which if coming from Lyon is Geneva.
You can buy Swiss passes at Geneva station, or online in advance from the Swiss Railways site here (self print): https://www.sbb.ch/en/leisure-holidays/travel-in-switzerland/international-guests/swiss-travel-pass.html

Buying a Swiss Pass in advance has the advantage you don't waste time at Geneva station, just catch the next train (or in some cases simply stay on the same train).

You can buy a ticket from Lyon to Geneva here: https://www.trainline.eu/

I don't think any trains from Lyon Part Dieu go beyond Geneva. All of the direct trains (except 1) are TER trains and they depart Lyon Part Dieu every 2 hours at 38 past the even hour and take 1 hour a 51 minutes to get to Geneva. The price is always the same, 29 EUR. Except if you are 60 or older, or 25 or younger, then you get a 25% discount traveling off-peak. Off peak times can vary. Week days everything except the 6:38 departure is off peak. On Sundays, everything except the 16:38 and 18:38 departures are off peak. Friday afternoon, most trains are not off peak. There is reduced service on Saturdays, but all off peak.

That lone non-TER is a TGV at 15:32 and costs a lot more and is no faster than a TER.

So you can just buy tickets at the station. No advantage to buying ahead. No seat reservations.

Thanks a million. Your advice is really appreciated. It's nice to know that there's no advantage to buying ahead. I'm planning on buying the Swiss Travel Pass for 8 days. If I understand correctly my children (ages 12 and 15) will travel free with me. Is 1st class worth the extra cost? It's around 200 CHF more, but I'm divided by 8 days and the fact that my children will be free, it might be worth the extra money. I'll be using the pass to travel from Lausanne to Zermatt to Wengen to Bern to Lucerne to Basel. Thank you.

You need to get a "Swiss Family Card" for your children. Quote from SBB website ( https://www.myswitzerland.com/en-ch/transport-reisen/billette/tickets-oeffentlicher-verkeher/swiss-family-card.html ):

Children under age 16 travel nationwide on public transport free of charge when accompanied by at least one parent holding a Swiss Travel System pass or ticket. The complimentary Swiss Family Card is available on request from all Swiss Travel System points-of-sale worldwide and within Switzerland.

I would say 1st class is not worth the extra. You get 2+1 seating (versus 2+2 in 2nd class) and proportionally extra legroom. But 2nd class is perfectly adequate. Seats are often 2-facing-2, with a small table in between. Very convenient for a group, you have your own "private" space.
See the "Man in Seat 61" page on Swiss trains. About half way down he has some photos, so you can compare: https://www.seat61.com/Swiss-trains.htm

1) "second class (? - whatever it's called) is great and comfy. Can be crowded at a prime time. Last year we visited a friend that married a Swiss fellow. We used the same pass as you're planning. A car would would been very tempting except that the Swiss friends don't have a car (the husband has a pass and the wife and kids buy as-needed) and we traveled together on the trains. (Yes, parking is a pain, but ...)
2) the android app for train schedules and which platform to go to, etc. was G.R.E.A.T. - it would set a standard for all else (though I've limited experience in passes).given that we had a pass, we could just a) review the train options as we sat at lunch and b) hop it easily. c) also, at rush hour travel time, we flowed easily w/ the crowd.