Redwood Regional Park is a good choice then. It's not a far drive from Highway 580 or Highway 13 but it's a forest of redwood trees, a trail network with something for everyone, and a wide open space perfect for exploring.
Stream Trail heading south.
There are two main trailheads, Skyline Gate on Skyline Drive and Canyon Meadow Staging Area of Redwood Road. I love running on the East Ridge trail from the Skyline Gate trailhead. But with kids it's the Redwood Gate where I go.
Drive to the end of the driveway and park at the end. There's a large meadow there perfect for picnics. Walk up the paved path up Stream Trail through redwood trees, meadows, and open spaces. There is a small and rustic playground about 1/2 mile up the trail which is both fun but also worth passing up if your kids will let you. There are more fun things to do up trail.
Stream Trail heading north.
You could walk for miles but only go as far and as fast as those little legs want to go. It's a great place to explore in the dry summer (read: shade) and wet winter with slugs, mushrooms, and flowing creek. Speaking of the creek, most of it isn't accessible to us to explore because they're restoring it and keeping it clean. But if you make it up to where Prince Trail meets the Stream Trail, keep going another 100 yards and you'll find an excellent place to wade, throw sticks, and do whatever things kids like to do when they're near water.
About a mile up Stream Trail, and totally worth it.
This is a favorite place. It's quiet, spacious and got a lot to offer. It's close to city and very far away. And we go all the time.