Hm. History.
The East Coast is pretty good for history, bearing in mind that most of it is focused squarely on the colonial era and revolutionary war. I don't know many of the specifics around New England, though I'm sure you'll want to see Boston and New York City, and maybe some of the old colleges and pilgrim stuff. I've always wanted to hit Providence, Rhode Island, too, if only to visit HP Lovecraft's grave and see the sites that inspired him.
Down towards DC, we have a thriving sight-seeing culture--again, focused on Revolutionary War. There's a bunch of civil war stuff in Gettysburg, PA as well, if you care about that sort of thing. In DC, there are the monuments, most of which you can hit with a quick tour on the National Mall. TONS of museums, courtesy of the Smithsonian, including the Holocaust museum for some Europe-relevant stuff. If you're really curious about America, we do have an American History museum and a American Indian museum, both of which I doubt have analogues over in Europe. They're also, like all Smithsonian things, free.
I think you'd personally dig a tour through the Masonic Temple in Alexandria, VA, too. There's lots of old Knights Templar stuff in there and some very cool architecture. For outdoors stuff, there are a few Chesapeake tours and on your way west a bit, there are some neat sites over in Shenandoah, including the fantastic Luray Caverns. Harper's Ferry, also out that way, is a neat place for more civil war stuff and old Western expansion.
Further on south, the East Coast gets less interesting, to me. It does change in tone very distinctly at about Richmond, and suddenly you're in the South. I don't know many sites to see in North Carolina barring the beautiful Smokey Mountains and accompanying Cherokee Reservation, but South Carolina has Fort Sumter, another civil war site (where open conflict began, in fact), and... probably some other stuff. I avoid the South. Don't know much about Georgia, either.
Then you get into Florida, where you can break into a few more American Indian sites with the reservations down there (Seminole Tribe), not to mention old Spanish colonies like St. Augustine. Miami is usually nice during Summer, as well (though so is Virginia Beach in Norfolk, so you kind of have your pick, not to mention a whole number of others).
And that covers what I know of the East Coast, which could keep you busy for weeks all on its own. The midwest, frankly, is pretty boring--but you can cover it with a drive through Kansas or Ohio or something, if you really want to. You could do it on the way up to Chicago, which is a hell of a city in its own right, and right on one of the Great Lakes. But that takes you a considerable distance from places like New Orleans (which you really should visit) and St. Louis.
It'd be worth your time to visit one or two of the national parks, too. Yosemite and Yellowstone are classics, but there are many more, including the Grand Canyon (which has a cool American Indian-funded glass bridge over part of it, for truluy stellar site-seeing). Rojito's right about Mount Rushmore, which you can gaze at in awe at America's arrogance in erecting manuments on sacred Indian land.
Ah, our ignorant ancestors... Lots of Indian Reservations in South Dakota, if you're into that.
I'm sure KMan and company would love for you to swing by Colorado, and Denver is a pretty nice area. I'm sure they know more about stuff there than I do. The Rockies, of course, are near there. A lot of people like to go up Pike's Peak.
The West Coast has a unique flavor, too, even north-to-south. I personally loved Portland and Seattle for their very unique feeling, and the Cascades are definitely worth seeing, out East of there. Mt. Hood is up in that area, too, I believe--if you dig hiking/mountain climbing.
California is very different, but I've only been to LA, which was sort of unexciting. I've heard good things about San Francisco. The lowest point in the US is out there, Death Valley, if you can stomach going there in the Summer.
There are Redwoods, too, and Rojito mentioned nearby (relatively) Las Vegas, Nevada, also in the region.
I'd just avoid Texas entirely, unless you have a burning desire to see the Alamo (neat battle site). But I'm biased. They are a unique subculture out there, so maybe it's worth it to you.
Phew. It's honestly a very big country, Vemu, so trying to hit all or even most of the above will probably break you. But I applaud your ambition. Just be sure to swing by and visit all of us! I'll hit some monuments with you when you're in DC.
Unfortunately, my place is too small for much crash space, unless you guys like sleeping on the floor.