One of my very favorite National Parks is Pictured Rocks National Lakeshore in the Upper Peninsula of Michigan. I love camping at Twelvemile Beach and going on day hikes from there, or just walking along the long beach. It's beautiful, so pristine.

www.nps.gov/piro/index.htm

June is a wonderful time there. There are wildflowers and the bugs are not bad yet. But the water is still pretty cold. When I've been there in July both the mosquitos, and worse, biting flies were plentiful. On the beach you could be free of them, at least. And in July Lake Superior is warm enough to swim in. But I prefer June.

Key West is a good start. There are some great camp grounds in the southern most tip of the US cross the 7 mile bridge. Then I suggest the Painted Desert and the Grand Canyon. Lots of stuff going through that way. Hike to the bottom of the Grand Canyon, it is worth seeing but do it during the full moon.
The Red Woods in California, if the parks still open. They are talking about closing it. Then up the coast. The Oregon coast is very nice if you get lucky with the weather. Creator Lake in Oregon is worth the trip. Wizard mountain. There is also lots of stops in through there.
Glacier National Park and Yellow Stone in Montana, Wyoming. Lots of mountain passes like Giants Ridge. Really fun area to go off roading and looking for geodes. Pikes Peak. Devils Tower.
In South Dakota the Black Hills are worth the stop, Spearfish canyon, Badlands. Northern Minnesota, the Gunflint Trail. Lots of bears. The Lake Superior shore line. To name off a few sights worth seeing. Haven't camped the NE so that's where my advice ends.
This list would take me 12-17 weeks. I have easily spent 6 weeks in Yellow Stone alone. In fact I take 3 weeks a summer just to travel and Florida was one trip, Grand Canyon was another. Then the Red Woods and Oregon was one, Yellow Stone and Glacier, then I went back to Yellow Stone again it was so amazing. The only grizzly I've ever seen was in glacier. If your doing this maybe just plan on spending a week in the Black Hills and a few days in the Badlands. Then get lost on the Gunflint, portage out of the US, due Lake of The Woods and go across through Canada back to the east shore.