Henry Coe is a good spot close to San Francisco for backpacking trips, especially for beginner backpackers. There are no reservations needed, just a self registration slot at the trailhead. With no bears plus some nice lakes to camp alongside, this spot is definitely a great place to try.
The park is particularly nice to hike through during the winter season because it is generally warmer than the San Francisco Area. Waiting to venture out too late into the summertime, however, can start to get uncomfortably hot as there are not many shady spots to hide you from the sun. Keep in mind that from Wintertime until April, the hills there are super green and particularly scenic.
The only con that you’ll need to plan ahead for is that there is not much available water. To my knowledge there is one tap (of non-drinking water) at the horse camp, but that’s it, so you will have to bring lots of water and pills to purify any water that is collected along the way.