We will be staying in Geneva for a couple of nights before driving to Annecy and on to Strasbourg. We will pick up our car on the French side of the airport. What is the best way to get there since we will be carrying 2 pieces of luggage each.
There's a train from the Geneva train station to the airport, which is free (or was when we were last there) if you get a pass when you arrive in Geneva. Or you can take a taxi. Have you rental voucher or other proof of car reservation, since you'll probably need it to get to the French side.
I know you have had lots of advice about not using the car for this trip on your other thread, but since it appears that you will, there is one more thing to caution you about.
I'm surprised that nobody mentioned it in your other thread but if your driving licenses are not from a country in the EU (you don't mention your residence in your profile) all drivers who will drive in France will require either an official translation into French or (much simpler) an IDP - International Driver's Permit. These can easily be obtained at AAA/CAA in North America. The IDP must be issued by an approved agency and in North America that is the AAA in the US and CAA in Canada. If you are in India it is a complicated process. If you are in Australia or New Zealand I'm afraid I don't know the competent agency. In the US it runs a little above or below $20 US. It may not be requested to be seen by the kid behind the hire car desk but it is a legal requirement and should be shown with your home license to police officer or to a person involved in a collision with your car.
I would have thought the easiest way to Strasbourg from Geneve would be Geneve > Lausanne > Bern > Basel > Mulhouse > Colmar > Strasbourg. That would run somewhere around 4 hours, but does involve beautiful Swiss Scenery and roads and will require a CHF 40 Vignette.
To avoid Switzerland would take the best part of two or more hours longer (as much as 10 hours in total if you avoid French tolls) and would take fuel and French tolls which would cost more than the CHF 40 cost of the vignette.
I did this a few years ago and found it confusing, so here are some specifics. Once you arrive at the airport, go inside the airport (Swiss side) and upstairs to departures. Facing the check-in desks, your back to the doors/street, go to your right all the way to the end of the terminal. The entry to France will be there, with a security checkpoint. This is what confused me. I thought it was security to get to the gates for air departure. I turned around and walked away a few times, and kept getting directed back to the same door. Finally braced up and went through. Lo and behold, it's a whole separate terminal. Just follow the obvious signs to car rental desks.
Part 2 - are you also returning it to the French side of Geneva airport? That is even worse. We could not find the tiny side road in the early morning darkness. Drove back and forth through the former border control (unmanned today) at least three times. Missed our flight home to Canada that morning. If you are returning there, be very sure you can recognize the turnoff onto a tiny road at a very sharp angle behind a barbed wire fence. Moral: return it in daylight and stay at a nearby airport hotel with shuttle service for the next morning.
We rented a car from the French side 4 years ago -- it is not all that confusing to pick up in the airport BUT it is very confusing to drop the rental back to the French side for your return flight home. The Hertz rental did give us directions but you have to be VERY aware of where you are so you can remember how to return. The signs for the airport will all be for the Swiss side and not the French side. It is completely doable you just have to pay attention and allow yourself extra time to return. I would not recommend an early flight. We had a 2 pm flight and therefore we returned in the daylight.