There most certainly IS camping in that region, Oss - but you have to book it in advance if you want a weekend date, as camping is mega popular out here.
FD, you mentioned The Redwood Forests as something you want to see. There are a couple of good choices for this that I know of -
If you want to just walk around some gorgeous and huge trees, in sort of a day-trip controlled environment, then I would highly recommend Muir woods, in Marin county, just across the Golden Gate bridge in San Francisco. Wow. What a great time you and your kids would have there!
If you are looking for a more rugged experience, with camping and hiking, try Humbolt State Redwood park, which is about 4 hours up the 101 (beautiful drive!) or maybe National Redwood park, which is another 30 minutes further but is supposed to be awesome.
Here's a pic of the size of trees you will see - with my fiancee for scale:
Those are nowhere near
the biggest ones. Not even close.
Sounds like you are going to have a great time. If you can get to see the Redwoods, do it - I can't describe their grandeur to you. Truly mind-blowing.