E.C.
google "Free museum days San Francisco" to find out which museums offer free admission on certain days of the month (If I remember correctly, SFMOMA and the DeYoung have free admission on the first tuesday of the month). The SFMOMA web site also has a great interactive called "The Country Dog Gentlemen Travel to Extraordinary Worlds" http://countrydogs.sfmoma.org/ - in which the two painted canines serve as "tour guides" through a couple of pieces from the museum's collection in a fun and approachable way - as well as giving kids the opportunity to electronically create their own works in the style of the highlighted pieces using a drag-and-drop menu.
Parks and Rec departments can also be a great resource for a variety of courses. If you don't mind travelling across the bay on weekends, the East Bay Regional Park district has some wonderful science and nature programs at places like Ardenwood Historic Farm and Coyote Hills Regional Park in Fremont, Crab Cove in Alameda, and Tilden Regional Park in Berkeley - some for a small fee and others free, many on weekends. http://www.ebparks.org/files/RIN_July-August_2011-EBRPD.pdf
For music - many youth orchestras produce excellent concerts for nominal admission fees (sometimes just requested donations) and it can be a cool experience for grade schoolers to see musicians that are not much older than they are. Summertime means that various music and theater groups may be offering free outdoor performances - here's the link to the free Stern Grove concerts for this summer http://www.sterngrove.org/season2011.html
hope that helps! check your library too - ours is a great resource for finding enriching and budget-friendly activities for our family
Edited to add: ranger programs at national parks and national historic sites are also a great resource to learn more about nature and history - the John Muir House in Martinez and the "Earthquake Tour" at the Point Reyes National Seashore are a few we've enjoyed recently. State parks too - a couple weekends ago we visited the Railroad Museum in Sacramento and the docents there are very helpful and the tour was very informative.