I loved camping on Baker Lake as a kid. We use to go with a group of families and take the wake board and water ski all day.
My parents own in a private camp ground right next to Rasar State park near Concrete up Hwy 20. Always a nice place near some great hiking.
Baker Lake is great, esp. if you have a boat. You can motor about 3/4 the way down the lake on the east side for some amazing beach sites that provide the most amazing views of Mt. Baker in the morning.
Fishing can be fun for the family but dont count on the big guys too much though.
Greenwater and Elbe hills are probably some of the best places I've been. I often camp in capitol forest as it's such a short drive from Olympia I can just decide to camp without planning and go from decision to camp set and relaxing in just over an hour including loading up my gear and stopping by the local grocery on the way.
Our family regularly visits two locations. Near Mossyrock, WA is Ike Kinswa State Park. And Camp Cushman, formally Lake Cushman State Park. Of course, I have two daughters, and that limits our camping experiences to places with showers. But if that's what you are looking for, I'd recommend both locations.
It depends on what I am in the mood for. If I want solitude and good exercise hiking, then nothing beats the North Cascades west of Mazama I almost always camp at the same campsite- especially early or late in the season.
If I am inetrested in wooded areas, I generally go to some part of the Olympic National Forest. I have camped in about 4 or 5 different sites and they have all been really nice.
Hiking Beacon Rock is one of those things that you should do. It is the world's second largest monolith!
Other than that, we have camped several state parks- none of them really stand out. I have also camped out at the Dark Divide which is truly primitive camping- no water, no toilets, not even marked tent sites!