I would suggest looking at the Point Reyes National Seashore, close to Petaluma, less than a hour north of San Francisco. It is a beautiful area ranging from thick woods to large open bluffs to the seashore. Because it is on the coast, the weather is much more temperate than up in the mountains. There are more miles of trails than you could explore in one trip.
Camping is only allowed in four designated campgrounds. They all require a hike to get there, and they must be reserved well in advance before they fill up. This area takes its reservations (and other rules) seriously, so I wouldn't try to wing it. I expect they have to be strict or they would be overrun, being so close to the SF area.
When we were there, about 15 years ago, we stayed for three nights in the Wildcat Camp. It was about 6 miles in to the campsite. I'm sure that you could coordinate a trip that would take you to different sites over multiple days.
One word of warning: the deer ticks were prevalent. Perhaps they won't be in early April, but they certainly were in August :)