Lyon to Geneva

I'm not sure which city you are asking about....I think Geneva but you put this in the "France" section.

If Geneva, the airport train station is located below, and to one side, of the main terminal building.

There are public trams that go from the airport into Geneva. You can see a map online.

Your hotel can issue you passes for free transport on the Geneval public transit system.

http://www.geneva.info/transport/card/

Looking at maps it appears the train station is located in the
airport,

There is a station (Gare de Genève-Aéroport) at the airport, the main station in Geneva (Gare de Genève-Cornavin) is in the city.

but trying to figure out how easy it is to navigate from the station
to the airport to pick up a shuttle bus to our hotel.

The airport station is as mentioned at the airport, just follow the signs and you are at the terminal.

We are staying at the Hilton next to the airport. I see that's a bit
aways from the Old Town area where we will want to go.

It certainly is, have you considered staying in the city instead?

The regular train from Lyon Part Dieu station goes to the main Geneva Station in central Geneva, often referred to as Geneve-Cornavin. Since your hotel is at the airport, you would change there to one of the frequent trains to the airport. From the airport station you could walk about 1/2 mile to the hotel or take the free hotel airport shuttle bus.

To visit central Geneva, you could walk or shuttle back to the airport train station for the short ride to central Geneva, or take a city bus from near the hotel. I guess I would inquire about this option from the hotel. Also, check if the Geneva airport hotel offers a guest card that allows free use of public transportation in the city.

No need to take a train from the airport rail station into Geneva. City trams/buses run from right outside the terminal and with the tourist card it should be free.

I agree with Badger, consider staying in the city itself.

If you're just spending one night in Geneva before leaving to the US, then maybe taking the Covid test in Lyon will be enough in terms of timing? I don't know how old the test needs to be. Suggesting that because rapid testing in France is plentiful and much cheaper than in Switzerland!

The test for coming back to the States has to be taken within the last three days before departure. Good thinking and a great tip, Balso.

We are doing the same thing in September, staying in Lyon, then train to Geneva and staying at the Hilton overnight for an early departure from GVA. I assumed the train takes us to the airport train station and it's easy to get upstairs to the airport where we can catch a hotel shuttle to the Hilton, according to their website. Depending on our situation, we might get our covid tests there at GVA if we cannot get it done in Lyon in time. I am encouraged by the posts here that say there are plentiful test sites in France! (I understand there is also the option of purchasing a test kit in the US and doing it ourselves). If we have time we will maybe do a quick bus tour of Geneva, but if not we can chill at the hotel because we will have done lots of sightseeing in Lyon, Alps, etc. I am really depending on that hotel shuttle because we have a 7:30 am departure and I hope they can get us there early the next morning.