If you have a car, I cannot stress going to Muir Woods and Point Reyes enough for a day (granted if you have the time too). It's so spectacular, and really only a half to an hour away from San Francisco.
At Fisherman's Wharf there is an extremely old arcade. I forget the name of it (it's a French sort of name like Arcade Fantastique or something) but it has a ton of super old arcade "games" that actually work, many dating from the turn of the 20th century. Most are games that are sort of like pinball but not quite (like a baseball game where you have a button that hits a steel ball with a metal bat into a field where there are holes and such for outs or hits). Many are not games but more like a crappy sort of animated show, but they are all super unique and there's no admission to the place or anything, just the cost of the "games" if you want to play them. I loved the place more when it was at the Cliff House, but the Wharf is a much bigger tourist destination so I can see why they moved there.
And for another you need a car for trip, a trip to Santa Cruz is not bad at all. It is home of the last Pacific coast amusement park (I could be wrong about it being the last, but it's one of the very few at the least), and the Giant Dipper roller coaster there is my personal favorite out of any one I've been on. It also has a nice public pier with shops and stuff. It's also kind of nice in the sense that you can just buy tickets for the rides you want to go on. Monterey is near by which has nice scenic drives (especially by Pebble Beach). I'm not sure I'd recommend it with all the other stuff to do in the Bay Area, but I personally enjoy it a lot.
At Golden Gate park the quintessential tourist thing to do is to go to the Japanese Tea Garden. Amazingly enough I had never been in all the years I grew up in the Bay Area, but I did go on a visit a couple years ago. While it's a bit overpriced (like $5 or $7 to get in) I will say it's a neat thing to see. It's the sort of thing I wish I could do in my backyard, on a much larger scale. I think I even shelled out the extra $4 for tea and cookies at the end.