Camping in San Francisco - we hate to break it to you that there aren't any campgrounds in San Francisco proper. You'll have to set up your tent further down the peninsula or across the Golden Gate Bridge in Marin County.
Also, you'll have to plan far, far in advance if you want to camp anywhere near the Bay Area on a Friday or Saturday night, especially in summer. There are a lot of campgrounds, but there are also a lot of people in the Bay Area!
We recommend camping out on Angel Island in San Francisco Bay for the neat experience - you take a ferry to get there! Bring everything you need, including your bike if you can (to avoid the overpriced rental bikes).
In Marin, Samuel P. Taylor State Park is tucked away in the redwoods, but the camping sites are close together and a little run down. It's a great location, and very popular with families, as are most campgrounds in the area. China Camp State Park is an interesting alternative - by the bay, it's not the classic camping under the redwoods experience.
Further down the peninsula from San Francisco, there aren't a lot of options until you get to the mid-peninsula, where San Mateo County runs the 152-site Memorial Park, with sites in the redwoods and a swimming hole in Pescadero Creek. On the coast, where Pescadero Creek enters the Pacific Ocean, the private resort Costanoa has tent camping, in addition to tent cabins you can rent and a lodge.
Closer to Santa Cruz, there are several state parks in the redwoods - Big Basin State Park being the nicest, with waterfalls, trails, and shaded, quiet sites.
If you want something a little more out of the way, check out Kirby Cove in the Marin Headlands. Only a handful of sites, and you'll need to hike in if you want to check it out without camping (campers get a gate code) Reservations are hard to get, but you'll get amazing views of the Golden Gate Bridge and the city.