In All About Eve, several scenes take place in the John Golden Theatre. Though the exteriors of the Golden Theatre were shot in the actual place, they used a different location for the interiors. This 1922 Theatre of San Francisco is still a popular venue for big musicals and other spectacles.
If you walk or ride through golden gate park heading west this is where you'll end up. A long expanse of beach, great place for a walk any time of day. Close to the sutro bath houses and the cliff house restaurant.
When the tide is low be on the lookout for the wooden remains of an old ship wreck that's right there on the beach.
A strange arcade and mechanical museum that used to be housed in the Cliff Houses near Ocean Beach but recently moved to the more touristy area of Pier 45.
It's filled with old coin machines that are really strange and all in working condition. Fortune tellers, black and white photo booth, boxers; I even saw one of a beheading. Just pop in a quarter (25 cents) and enjoy.
The ferry that takes you to Alcatraz Island leaves from Pier 33 in the Fisherman's Warf area on the embarcadero. It is a little expensive to go out there, but the tour of the prison and the views from the middle of the bay are well worth it. Plan for it to take at least 3 hours, but you can stay longer if you want. They also have a night tour of the island, which I haven't tried but sounds amazing.
You can buy tickets in advance:
Icon of SF, this 1937 suspension bridge is well worth a visit (the best way to see it is on bike, you can ride all the way across, taking the bike path along the bay from Fisherman's Wharf and then either ride back or take the ferry from Sausalito).
On the darker side, it's worth knowing that this is one of the number one sites in the world for suicide. There is an interesting documentary about the phenomenon called The Bridge by Eric Steel.