Okay, a few ideas…
First of all, fly over it! (Wx permitting). Class B floor there is 3000’ so an overflight at 2000’ works well. Also a great way to see Alcatraz, North Beach area. For ground level:
Take BART to Embarcadero station. From there, it’s a scenic 6 mile walk to the south end of the GG bridge. Or grab an Uber, or, if adventurous, try to figure out the bus system. For airports, Concord (KCCR) or Oakland (KOAK) are within walking distance of a station; from Livermore (KLVK) it’s a short Uber ride (or again, try navigating the bus system) to the BART station.
Driving: KLVK has Enterprise, call ahead to reserve. KCCR also has rental cars.
Note KOAK is actually two airports: south field, for commercial traffic, and north field, for GA. Separate control towers. I don’t actually know what the walk is between them, maybe 2 miles? Of course south field has lots of rental cars, train shuttle to BART, etc. I’m not sure of the rental car situation at north field but I think there must be some.
Notes:
All bay bridges are toll bridges, and they’re all automated. Make sure your rental car comes with a bridge transponder (extra cost, of course). Stay off the roads before 9 am, and again between 4 pm - 6 pm. You won’t like the traffic.
Wx: airports near the bay/ocean are subject to low costal stratus, especially in the morning. KCCR and KLVK are usually VFR.
My actual recommendation: I’d try for KCCR, rent a car. If staying overnight, Sheraton on the field. Second choice would be LVK and Enterprise. This assumes you want to drive over the bridge. If you just wanted to see it, I’d take BART from CCR and walk/Uber to the bridge. That’s a bit more adventurous!
Edit added: if driving north across the GG bridge, take the first exit on the Marin side, follow your map west and up. There is an abandoned Nike missile site on the Marin headlands which offers great views (wx permitting) of San Francisco.
Second edit. Heading into San Francisco bring a sweater or jacket. The City can be surprisingly cool, even in the summer.