GO ON THURSDAY NIGHTS FOR A DISCOUNTED $5 TICKET! This museum, right in the heart of San Francisco's Civic Center, has an amazing collection of Asian art. Not only that, but the exhibit I saw there, called Gorgeous, was pretty much the best curated exhibit I've seen in a long time. The range of pieces they brought in was impressive, but more than that, instead of yawning/snorting at the pretentiousness of the placards next to these pieces, I found myself really appreciating the additional context without having this generic art historian telling me what to believe. Strongly recommend.
This museum has a good collection of Chinese, Japanese southeast asian, tibetan, korean and Persian artefacts. Don't miss the 28 Chinese exhibition (summer 2015) showcasing contemporary art from well regarded Chinese artists. The museum has an in-house cafe and a really nice shop with well curated artisitic gifts.