We're also an outdoor/camping family! I also have a ton of experience -- everything from a month backcountry to car camping.
Last summer, I spent a week with my then 3 year old in Big Bear. We had a great time! The campground is really nice, family friendly, and running water (which is nice with a little one). It was close enough to Big Bear that we could get town, but not in the middle of town. Bring bikes! There is a nice visitor center not far, and a fun bike path.
We went in the middle of the week, and it was not too crowded. The weekend it was full, but I think it will be full anywhere!
The other thing to remember is that car camping means full on luxury. Bring a blow up mattress, folding chairs, two burner stove -- all the stuff you don't bring when you backpack.
We've also done Leo Carrillo (crowded, but awesome b/c it's so close to the beach), Joshua Tree (great, but too hot in summer and no water) and Yosemite (probably too late to get a campsite). I've heard good things about Sequoia as well.
My general feeling is that if you love it, your kids will love it. Have a great time!