Ride a cable car. It’s not likely to be the quickest way from point A to B, but way more entertaining than the buses back home.
Giant trees are well worthwhile, even if it’s just a quick look. Your itinerary likely precludes going all the way up to the Eureka area. Muir Woods, however, is right across the Golden Gate Bridge in Marin County. Big Basin Redwoods SP is about 1.5 hours SW of the airport, but that would eat up most of a day, and there’s not a whole lot else in that direction. If it was winter, you could combine it with the elephant seals at Ano Nuevo, which are awesome.
Fisherman’s Wharf, Ghiardelli Square, and Pier 39 are touristy (some would say tourist traps), of the Hard Rock Cafe/saltwater taffy variety. Nothing you couldn’t see in a dozen other cities. Some people love that sort of thing, some hate it.
Haight Ashbury has some nice old Victorian houses and lots of head shops, but the counterculture angle is a little dated and a bit of a put-on. If you are into the whole summer of love historical thing, certainly worth it. Otherwise, you could see most of the same stuff at home.
The Legion of Honor and the de Young are both art museums and usually have a few traveling exhibitions in additon to their fine collections.
In addition to the de Young and the tea garden, Golden Gate Park is home to the California Academy of Sciences. Packed full of screaming kids, right? Yes, but on Thursday nights it’s 21+, with live music and cash bar. If drinking in a natural history museum/aquarium/giant terrarium sounds like fun (of course it does), it’s the only place in town to do it.
Like raw oysters? Try Hog Island Oyster Bar, in the Ferry Building and also in Oxbow Market in Napa. Don’t like raw oysters? They have baked oysters and ambrosiacal clam chowder. Don’t like oysters? That’s what I thought, until I ate some of theirs. Totally different than Gulf and East Coast oysters.
Like dim sum? Go to Yank Sing. It’s a bit more expensive than average, but the food is quite a bit better than average, so the extra money is well spent. Get some xiaolongbao.
Union Square and it’s environs is nice enough, but the shopping is probably nothing you couldn’t do at home.
The coastal environment is very different than the east coast. Point Reyes National Seashore is not far from where you will be. More or less any section of Highway 1 is packed with gorgeous views and plenty of places to stop and look. You may as well make use of that rental car.