Flying in to Geneva...Where to GO? what to DO?

Head to Chamonix. You can go by bus very inexpensively, and there is no need for a car once you get there. Lots of hiking and other adventure options, and it is generally less expensive than Switzerland for meals and lodging.

Don't stay in Geneva. For us, it was more expensive than the other places we've been to in Switzerland. Chamonix is a good idea. Another one would be Luzern. Accommodations might be hard to find in Luzern at this point. There are small communities on the lake that might be available and a lovely place to stay.

I used to live in Geneva, and while I understand that it's probably not worth a chapter in a Steves' book, there are still things to do there. Also, there is cheap accommodation at www.cityhostel.ch which is about 3 blocks from the train station.
I'll just tell you what I'd do in your situation.
Arrive Geneva, and just before you exit the baggage area, pick up the free 80 min transportation ticket from the machine. Then you can take either the #10 bus on your left as you exit the airport, or go down into the underground mall (further left) and catch the train. Both take you to the main train station, Cornavin. I'd book one night at City Hostel so you can undo some jet lag and see a bit of the city.

If the weather is good you can take a free bus (included in your hostel/hotel stay) over to Calvin's Church and climb the bell tower, rent free bikes (www.geneveroule.ch), visit CERN. Not great weather you can go to the Red Cross Museum or go shopping in the underground mall near the train station. (Left out the main entrance, cross one street, look across the other to see the McDonalds, enter it and go thru it and you are at the top level of the mall.)

Book train tickets to Lauterbrunnen which is about 3.5 hr trip. You can get inexpensive rooms there at Valley Hostel, or if you want REALLY cheap, bring your own pup tent and stay at www.campingjungfrau.swiss . If you buy your own food at the local stores, you can save a lot.
There's easily enough stuff to do in the area to keep you busy for 10 days, assuming you like walks with fantastic vistas.
Also, I wouldn't rent a car since it will just sit idle in the Lauterbrunnen/Murren area.

I'll send you a site with our dozen favorite hikes with maps and pics. Just click on your name in upper right to read your forum mail.
Consider buying a Berner Oberland Pass

https://www.regionalpass-berneroberland.ch/en/pass-and-infos/area-of-validity/

if you love using their cable cars and mountain trains, or a half fare card ($130) is another option.
You could also make a side trip or stay in Lucerne for a couple of days.

This is already amazing advice. Thank you so much. I don't know how to respond to individual people separately, so I apologize. Specifically, "Shoe," that is all amazing information. We have been to Lauterbrunnen and hiked from there to a couple of huts above Gimmelwald. It was amazing, and I would do that trip over exactly as we did. I think we'd like to try another area of Switzerland - will definitely do as you say in Geneva. I'd still love to see the hikes you recommend! Do you have other area advice as well? THANK YOU!

I am sorry Shoe, if I offended you. My first reaction when I saw that finances were an issue was that of all the places I went to in Switzerland, Geneva was by far the most expensive. I couldn't find a "value" place to stay and it was mid January. I stayed in a hostel with my two college aged daughters. It was fine and a good experience to have, but I'm feeling like I'm too old for that! Anyone interested in physics needs to absolutely go to CERN. Transportation is free, CERN is free. You just need to get CERN tickets in advance and the window is very small. I think it was two weeks and sometimes the tickets are gone in minutes. The tour was fantastic, my physics minded daughter was in heaven! They also have an above ground bike/walking train that follows the path of the collider, though it is closed in the winter. We intend to some day go back for that. We took a bus to the UN building. Unfortunately the Chinese president was there so the grounds were closed but it looked like it would be interesting.

If you like science you should try to snag a free tour of CERN while in Geneva. You have to sign up in advance and the slots go quickly.

You will also be close to Lausanne and Montreux. Fun cities and the lake is a great place to be on a hot August day.

I second the plug for Chamonix. Plus, you could take the Mont Blanc Skyway into Italy and Valle d'Aosta - if even a day trip. I haven't done that yet, but it looks fantastic! Seems like plenty to do between Geneva, Chamonix, and Lausanne-Montreux corridor! And, you could limit your travel expenses by remaining within that loop.

Zermatt and the Matterhorn are within striking distance from Montreux. It's a great location and fantastic hiking. Worth your consideration.

If you do happen to make it to Bern, float down the Aare river with the locals - that was a fantastic and unexpected cultural experience for me one August - even if my shorts were wet for a couple of hours after.

Also, keep in mind that Rick has a strong bias towards cute and quaint, and that's not Geneva. It doesn't mean it isn't worthy of a visit, however.

Here is the information on visiting CERN.

https://visit.cern/tours/guided-tours-individuals

Jules, I am not offended at all. Geneva IS expensive, though I found ways around it, such as shopping over the border in France.

Some things to consider in Switzerland near Geneva:

In Lausanne, I enjoyed the Olympic Museum and (especially) the Museum of Outsider Art. In Montreux, I enjoyed walking the waterfront promenade. I wasn't as taken with the Chateau de Chillon as most seem to be (and it was very crowded), but if you're more of a "castle person" than I am, you will want to see it. Most memorable of all was visiting Audrey Hepburn's grave in Tolochenaz, near Lausanne.