San Francisco Conservatory of Music Preparatory division. Also a great place to join an ensemble (opportunities for Adults as well)!
The conservatory also has the college age concerts and master classes, which are well worthwhile. Recitals are free and the audience is appreciated, orchestra concerts have great repertoire and young musicians.
The San Francisco Youth Orchestra is a fantastic youth orchestra to see.
More concerts: San Francisco Symphony, Opera, and Ballet of course! Look for rush and standing room tickets for cheaper options. The San Francisco Chamber Orchestra is fantastic, and always free! All groups also have children’s concerts. Crowden Music School also has great concerts that are geared towards kids.
There is also a fabulous resource from the San Francisco Classical Voice for families. They have a section for “kids and families” on their website, which will (among other things) allow you to subscribe to emails of classical music events that are kid friendly in the bay area. Find it here.