From 6,000 year old artifacts in the Asian Art Museum to claymations created today at Zeum, San Francisco's museums offer a diverse wealth of exhibits.
Art-wise, MOMA and the Contemporary Jewish Museum cut the edge of modernity, while the Legion of Honor and deYoung are primarily fine arts oriented (eg, Picasso, not iPhones).
World renowned for hands-on exhibits, the Exploratorium defines itself as "a museum of science, art and human perception". The California Academy of Science houses a rainforest, planetarium and multiple aquariums, and the nearby Conservatory of Flowers offers a tour de fleurs through an architecturally stunning Victorian greenhouse.
Outside, the Botanical Garden is home to immigrant and native plants in abundance, while the Zoo complements with fauna. And, of course, Alcatraz combines natural beauty and an infamous/former high-security "zoo" for humans.
Other/smaller historical museums are excellent - the Cable Car Museum runs the cables that run the cars, the Fire Department Museum chronicles an era when local firefighters were well-known heroes, and the Musée Mécanique offers a peak in the entertainment of the early 20th century - and the Walt Disney Family Museum chronicles the pioneers of animation that replaced it :)