While I agree with much of what Bart had to say, I will take a somewhat different tack. If the idea is to see the United States in a fashion that would allow you to be able to gain some insight into its breath and character, I suggest the following order and places to visit (though the actual time you spend in each location is, of course, up to you):

1. Land in Washington DC, spend a couple of days there, and then take the Route 95 corridor up through Baltimore, Philadelphia, New York, and Boston.

2. From Boston take any combination of Routes 93, 89, 91 or 87 north to Montreal, Canada.

3. From Montreal take Route 401 along the St. Lawrence River through Ottawa, Toronto, and finishing in Detroit.

4. In Detroit you have several choices; take a wonderful ferry ride up through Lake Huron and Lake Superior to Duluth, Minnesota, or drop down Route 94 and pick up the toll road Route 80 west to Chicago.

5. From either location (Duluth or Chicago) get to Route 90 west and head for Rapid City, South Dakota, Mount Rushmore, the Badlands and the Black Hills.

6. Continue on Route 90 (drop south for a day to Yellowstone National Park), then, as you enter the Rocky Mountain Range, continue through Montana (Little Bighorn) and Idaho (Bitterroot Range of the Northern Rockies).

7. Route 90 will take you through the Cascades to Seattle. From Seattle south you have a continual choice of Route 5 (for speed), which runs along the interior portions of Oregon and California, or Route 101, which runs along the West Coast all the way to the San Francisco Bay area.

8. From San Francisco, take Route 120 to Yosemite Park then across the Sierra Nevada's to Route 395, then west on Route 190 through Death Valley to Las Vegas, Hoover Dam, Grand Canyon, etc.

9. After the Grand Canyon you want to work towards Route 40 east, going through the Petrified Forest, then Albuquerque. Spend a day up to Santa Fe and Taos, and then pound through Texas and Oklahoma, Little Rock, Arkansas to Memphis, Tennessee.

10. Drop south on Route 55 to New Orleans, and when you're done, take Route 10 along the Gulf of Mexico to Florida.

I have been to every one of these places, as well as most of the United States. I took the time to pass this list to several of my colleges at work, who regularly travel all over the US, and they agreed that, given your time constraints, when you leave you will have a relatively good idea about America (and a portion of Canada). While some very nice places have been left out, you have to make choices. I've done these routes in various configurations over the years, and plan to take them again. I wish you Good Luck!