If you are at Golden Gate you want to drive across and go visit the vista point at the North end of the brigde during the early afternoon (I'm talking about the vista point that you reach by taking exit at Alexander Ave, not the one at the North edge of bridge). It gets very foggy late afternoon so you wont be able to see the bridge from there. It's a beautiful site with Pacific on your right and city/bridge on left. Golden Gate park has lots of things to see, if you are a museum lover, do check out the Asian Art museum. Every dollar you pay for ticket is worth it. Queen Wilhelmina park has lovely flowers and windmill which is right by the ocean.
Palace of Fine Arts is worth one visit. As suggested in previous post, Fisherman's wharf and Pier 39 are a must see. For Alcatraz you will have to reserve tickets about a month in advance. You can do this online. It's tough to get them right on spot. If you are a shopping person, China Town on Grant-Columbus streets is a must-visit place and union square - a shopper's paradise. Dont forget Ghirardelli square for chocolates and sundaes. Coit tower and Lombard (most crookedest street) between Hyde and Leavenworth. Sony's dolby theater near Market st. is something exciting and new but I havent been there so dont have much info on it.
If you are a hiker, well let me know I will give you list of hiking trails you can do. How much time have you got to spend in the city?
What do eat? Let me know what cuisines you like and I will suggest accordingly. There's very good Indian restaurants around the area too like Gaylord right off of Ghirardelli Sq. In China town you have world's famous garlic loving restaurant Stinking Rose - I have heard their food is very good and famous.
If you are a music buff who also likes American music, you want to check out Haight st at Ashbury. Better warn you, you will find some weirdos around there too so better just stay alert. Nothing scary sort but good to know of the area before being there. There are some good restaurants on Haight st.
A nice walk along on the walk way along Embarcadero is very pleasant too.
If you want more suggestions for rest of the bay area, just say it
I'd love to give more points of interest depending upon what you are interested in seeing.