Mount Rainier is one of my favorite National Parks. I've climbed the mountain, hiked around it, and skied it many times. August is definitely a very busy time there, and depending on how early in the month you arrive, it is possible that there will still be snow at Paradise and Sunrise (fun to watch kids sledding in snow wearing shorts and sandals when its 70 degrees out). Ohanapecosh, White River, and Cougar Rock are all great camp sites, however none have any good views of the mountain from the campsite. Ohanapecosh buts you right near some of the biggest trees you'll see outside of the great red woods. If you chose to stay at Sunrise or Paradise (where you get the great views) you'd be "camping" in the parking lot. If you're willing to drive a bit, Lake Mowich on the North end of the park will get you away from many of the bussed in tourists, but you'll need to drive quite a ways out of the park and then back in to get there, and it's all logging roads up to the lake.