Some ideas I've found: campgrounds in the Golden Gate National Recreation Area https://www.nps.gov/goga/planyourvisit/camping.htm. I've only camped there once, at Haypress (1 mile hike), but a reservation was easy to get a few weekends in advance in July this year. Coming up on winter time campground reservations get much easier to obtain (for example at Point Reyes, Mount Diablo, Anthony Chabot) but still not free. If you want something adventurous, the Juniper Campground on Mount Diablo gets quite cold! As for backpacking, Henry Coe is a great idea, as are some of the East Bay Regional Parks which offer inexpensive backcountry camp sites (not dispersed camping) https://www.ebparks.org/activities/camping/backpacking.htm. I've also found these easier to reserve than the usually campgrounds. Point Reyes too but it's hard to get a reservation until November. I found "One Night Wilderness: San Francisco Bay Area: Quick and Convenient Backpacking Trips within Two Hours of San Francisco" a useful book not only for places to backpack but also really detailed descriptions of the sites and trails.
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