It sounds like a grand trip!
What sorts of things are you interested in seeing and doing? What sorts of amenities do you look for when camping? Some general sorts of guidance (or even more specific ones) would be most helpful in providing useful suggestions.
A handful of general recommendations for most anyone would include The Henry Ford (outside of Detroit), Letchworth State Park (upstate NY), Watkins Glen State Park (also upstate NY), the Shelburne Museum (VT), the White Mountains of NH, Acadia NP (ME), and Skyline Drive and/or the Blue Ridge Parkway. Chicago, Boston, NYC, Philly, and especially Washington, DC all have some excellent city attractions, both historical and cultural.
I've stayed at Greenbelt NP outside of Washington and taken the Metro into the city from there; it's about a half hour walk (at a reasonably quick pace) from the campground to the College Park station from the D loop of the campground. If you don't mind dry camping and a little walking, it's an economical way to see the city.
Do be aware that many New England campgrounds close around the first half of October or so. Foliage tends to be the best (in Vermont, at least) around Columbus day or maybe a little bit before, though it varies from year to year, and progresses south as time goes on. Columbus day weekend is a busy tourist weekend; reservations for it are highly recommended. Labor day weekend is, of course, also a popular camping weekend.