So the SF pasttime isn't baseball or anything like that, it's laughing at tourists who wear shorts. All the visions of warm, beachy California are lies spread by the Los Angeles tv industry. SF can be cold and windy even in summer (actually especially in summer, the land warms up and causes the fog to roll in) and the fog is beautiful but it's FRICKIN COLD. I always bring a jacket to SF no matter what time of year it is. It can be 30 deg C maybe 20 miles away and 10 degrees C in SF. No joke.
SF has awesome food, with a lot of Asian and Mexican influence. Here's a great list from all price ranges: http://www.sfgate.com/food/top100/2014/
Pay particular attention to Mexican/Central and South American food, since I know the UK has great Asian food but not Mexican. The burrito is a Californian staple. Try the al pastor and the carnitas.
If you like dim sum, try Yank Sing on Spear St. It's good and they're used to tourists.
There's a number of smaller venues that are fun. The Fillmore is out of '60s SF and is just fun to hang out in. The Great American Music Hall is similar. There's lots of bars that have live music, too.
de Young Museum. London's museums are better but this is a gorgeous modern building.
Asian Art Museum. I forget exactly where this ended up (it moved) but it has an amazing collection if you like this kind of thing.
Exploratorium. It's a kids science museum, but it's super fun for adults too.
Fisherman's Wharf. It's a tourist trap and honestly nothing exciting. You can go look at sea lions, who like to sun themselves there. You can catch a boat here to go out on the bay, which is spectacular when it's sunny and boring if it's foggy.
Alcatraz. Everybody wants to go here. It's interesting, but it's a prison, so adjust your expectations accordingly. The views are amazing.
Golden Gate Bridge. You can walk across it (about a mile and a half I think) which is awesome when it's sunny. If it's foggy, you'll see nothing.
Powell St. This won't be funny to anybody who isn't in software but if you stand on Powell and Market and look up Powell, you're looking at approx. a billion startups.
Our public transit sucks compared to yours, and while getting around SF is pretty easy, getting out is difficult if you don't drive.
We have BART, which is okay if not stellar, and there's Muni which can be gross. Check out http://www.511.org/ for getting around, or just use Google Maps. I might just take taxis or Lyft. They're no london cabbies but it works.
Napa Valley. Napa is gorgeous. It's about an hour drive north of SF, and I know there's plenty of groups that will take you around for a day. Sonoma is prettier and cheaper imho if you have a choice between it and Napa. Don't fall for the Wine Train, you don't actually get off the train.
Yosemite. Yosemite is about a four to five hour drive from SF and there's no way to get there other than drive AFAIK. I honestly haven't been since I was a kid (need to fix that) but everybody I know says it's worth it.
Santa Cruz. Santa Cruz is about an hour and half drive south from SF and sort of the epitome of california hippie culture. It's a cute town with a lot of stuff to do, lots of bookstores and good food and all that. They have a real beach that you can walk on (the water will still be freezing cold) You can do this via public transit, I believe there a few busses that run between SF and Santa Cruz.
Tahoe. This is about a four to five hour drive and it's the state's main ski area. We've been in a drought for like four years so check the resort status if you actually want to go skiing.
Craig Kerstiens's list: http://www.craigkerstiens.com/about/travel_wine.html