Best route from Geneva to Salzburg

Is there some reason that you need a car? Train would be easy. Also, I must assume you got some kind of a special deal to fly into Geneva and your destination is Salzburg.

Angela, I would use Viamichelin: https://www.viamichelin.com/
This says 726 Km, 7h36, via Munich. That is non-stop driving time. As you are going to have to stop (fuel, eat), you need to add at least another 2 hours.
The route that gives is the quickest, keeping north of the Alps via Munich. I would suggest overnight in Innsbruck. This is a more southerly route, through the Alps all the way. It will be slower but a lot more scenic. You should make a hotel reservation

However, I would add some warnings:

• Is this a rental car, and are you picking it up in Geneva and returning it in Salzburg? If yes, are you aware of the surcharge for one-way international rental? Have you had a quote?
• If it is a Swiss car, it will already have a Swiss Autobahn toll Vignette. If not, 40 Franks.
• You will need an Austrian Autobahn toll Vignette if you are driving on any Austrian Autobahns.
• Depending on which country issued your Driving Licence (you don't say where you come from), you may need an International Driving Permit as well as your licence.

will you be doing this driving after a long flight, so you may be jetlagged?

I will be arriving in Geneva at 1 pm after a nighttime flight from the U.S. I hope to sleep on the plane.
I am picking up rental car in Geneva and returning rental car in Geneva (Swiss side). I do have an international driver's license.
Where do I acquire an Austrian Autobahn toll Vignette ?

Where do I acquire an Austrian Autobahn toll Vignette

At a fuel station on or before the border. Or a border post, but these only exist on the Swiss-Austrian border, not the German-Austrian border.