– walk up the steps on the hill to Coit Tower; views on the way up and in the parking lot are fantastic. Go up to the top of the tower for more views and the WPA murals. On the walk up the hill you will likely see wild parrots.
– take a walking tour of Chinatown, for the interesting back alleys
– Golden Gate park, especially the de Young museum. There are always interesting exhibitions and the de Young tower had spectacular views. On Friday nights, they have concerts in the lobby and in the bookstore in the tower. Also in Golden Gate Park, the Japanese Garden is great and it is right next to the de Young.
– if it is a sunny day, do not miss the camera obscura near the Cliff House. It is a building-sized camera that makes an image of the surrounding view. Well worth it!
– i agree with those who say to take the Alcatraz tour and to walk across the Golden Gate bridge.
– if you want to browse hipster shops, Valencia St in the Mission District is the place to go. Start at 926 Valencia (Dave Egger’s pirate shop) and go from there. There’s a great Indian dosa place called Dosa a few blocks away.
– go to the Legion of Honor museum (an art museum). The collection is just okay BUT the building is in a park with some of the most breathtaking views on earth. And if you will be there after January, they will have da Vinci’s Lady with a Unicorn on exhibit (on loan from Italy).
– The Ferry Building! And the farmer’s market in front of the Ferry Building!
– i believe you are a vegetarian, right? Green’s restaurant is an iconic vegetarian restaurant – upscale and delicious.
For nature:
– Muir Woods for hiking in redwoods
– Point Reyes National Seashore for beautiful walks/ hikes overlooking the gorgeous coastline;
– the cowgirl creamery store in Point Reyes Station for picnic supplies for your hikes;
– kayaking in Tomales Bay, with oysters to follow at the little oyster shacks.
– Stinson Beach for a funky surfer town and a great beach.
There are a million other suggestions I can give you! If you want to do an overnight away from the City, Big Sur, Monterrey, Point Lobos, Carmel are highly recommended but they are too far for a day trip.