All major agencies give you the choice of renting from the French or Swiss side. For example, Avis has 'Geneva Airport (French side)' and 'Geneva airport, Swiss side' on their website. Generally, rentals on the Swiss side are a good deal pricier than those on the French side, but they have a greater selection of automatic cars especially in smaller, non-premium car groups.
Just compare both options before booking.
The 'vignette' toll tag which you'll need to travel on Swiss highways and is included on all rental cars in Switzerland costs all of 40 CHF, in case you need to buy one when renting from the French side. This usually is more than offset by the difference in rental cost. Also, especially if you have a diesel car, fuel is cheaper in France, so fill up there before returning if you want to save a few bucks.
https://www.gva.ch/en/desktopdefault.aspx/tabid-72/ has all the details about how to rent a car at GVA.

If you want to drive into Italy, carefully read the fine print of your contract and/or research to verify before reserving/picking up. Some agencies limit which cars you may take into Italy - Avis doesn't let you drive there with Mercedes and BMW models, for example. France is generally OK, as is Germany.

It is pretty easy to find parking, but be prepared to pay dearly, especially in Geneva (also, if you stay at a hotel, check parking rates. Those can be very pricey there). Personally, I wouldn't bother with a car while staying in Geneva and only pick it up before leaving.
Switzerland generally has street parking regulated and limited. Check if there's a 'parking disk' in the car, because you'll need it to park in the ubiquitous 'blue zones'.