I think you should narrow down what you want to get involved in. Let me know, I have lived in SF for 5 years and I am very happy to help research or make introductions!
Ali Community Manager
In addition to what was mentioned, you can join find lots of ideas at churches and temples, on billboards at grocery stores and hospitals, and by talking to your neighbors.
Depending on your profession or sporting abilities there are tons of options there too. I'm a lawyer and can do legal aid clinics for folks, as a mediator I have mediated noise complaints between neighbors with Community Boards and if you're a bicyclist, I've helped the SF AIDS Foundation in the annual charity bike ride from here to LA, the AIDS LifeCycle.
And yes, sometimes there are wait lists to be volunteers at places too, but don't be discouraged!
If you mean your neighbors, one of the best is to start a neighborhood watch group so that you can all look out for each other. The benefits that come from just knowing your neighbors is incredible. Contact SF SAFE on how to start a group.
If you want to learn how to best survive the coming eathquake take the Neighborhood Emergency Response Training (NERT) provided free by the fire department. You will learn how to do triage first aid, rescue work, what supplies to have, where dangerous areas are, how buildings will behave and much more.
Volunteer for park clean-ups, organize any kind of clean and green event, throw a block party and the list goes on. Contact the Neighborhood Empowerment Network (NEN) or go to empowersf.org for more info and for lists of things going on in the many communities in SF.
There is also the Volunteer Center, the Interfaith Council and of course Homeless Connect.