What time of year are you going? In mid-July there could still be snow covering all the trails at Mt Rainier (no reason not to go though!).
I go to each of them 2-3 times a year. For me it's North Cascades (free!), but probably not great for kids at that age range. I mostly agree with the other comments, but don't worry about "doing them justice" - a couple days in a park is pretty good.
Rainier: Spend a few days. Paradise is most people's favorite and has a web of trails. If the wildflowers are blooming it's awesome. The Sunrise area is cool too. You'll probably also have fun driving the windy roads south of the mountain. Carbon river / Mowich entrances are not worth the time. I actually haven't camped at the drive-in campgrounds at Rainier.
North Cascades: At a minimum you could drive along highway 20. It is my favorite road. There are tons of pull-outs. If you're up for a stellar hike (and there's no snow), google pictures of Cascade Pass. Some people, for a scenic drive, go south after that to Wenatchee, then west on highway 2, but that's a long day and I don't think highway 2 adds much.
Olympics: It's all about the coast. The mountains aren't are interesting as NCNP/Rainier and they're not connected to the coast. The Olympic coast is perhaps the most untouched coast in the 48 states and is pretty rugged. Unfortunately, it's quite a drive to the coast - it's the only place I don't day-trip from Seattle.
Other things: St. Helens is interesting (where else can you see a huge mountain blown in half?), but a long drive. The view you see from the visitor center isn't as great as if you climb it. There's nothing else to do other than view the mountain since