Welcome aboard the RoadTrip America Forums!
To begin with, I don't know where you're starting or ending your trip. There would be one set of roads involved in driving from Duluth to San Francisco and an entirely different set involved in going from Milwaukee to San Diego. So without that basic piece of info, it's impossible to even recommend major attractions (Yellowstone or Bryce?) let alone specific campgrounds. But in general, I recommend looking at each state's park website for the facilities available at state campgrounds along your route. These generally tend to be a bit more available and often cheaper than national park or commercial sites, and you will also have a greater chance of meeting locals who can put you on to some lesser known attractions in the area. Other sources of camping are national forests and, especially in the west, Bureau of Land Management (BLM) acreage.
Also, for anyone to help you they would need to know how much time you have available. "End of May" to "Early June" could be as little as a few days (totally impossible, of course) to a week and half (doable, but no time for sightseeing) to 4 weeks (time to see not only the major sites, but some of the just-as-rewarding smaller venues).
AZBuck