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Sixteen days for the cross-continental-journey from New York to Seattle should be quite sufficient to let your kids get a taste of America. While both the Interstates and more local roads have their good and bad points, your trip should be long enough to take advantage of the best of both. Only by using the Interstates will you be able to make this journey in a relaxed and timely enough manner to see everything that you want. But don’t just slavishly use them as simple high speed routes to be followed hour after hour just to get somewhere. Instead, take advantage of the many small parks all along their lengths to show your children something of how Americans recreate other than just going to the big viewing sites.
Also, I appreciate your distinction between seeing the “personality” of America and what most people ask for, the “real” America. While the Interstates and the culture that thrives at their exit ramps are just as real as any other part of America, and they do reflect the personality of “
America as a whole, if you want to experience the different personalities of America, you’re going to have to occasionally get off these main roads and travel some of the smaller highways and stop actually in some of the smaller towns. A few that I can suggest for different segments of your drive include:
US-20 across New York State: largely follows the old Erie Canal with several historic sites
US-62 and US-71 through the Ozarks: scenic drives and some of the early cavalry forts on the plains
I-80/US-30 and US-385: The first two follow the old Oregon Trail and the third will bring you up into the Mount Rushmore area from the scenic south. Take advantage of some of the many other attractions around Mount Rushmore such as Badlands National Park, Wind and Jewel Caves, the Crazy Horse Memorial, and Devils Tower.
US-12 through Lolo Pass: Follows the route Lewis and Clark used in first exploring this section of the country. Rejoins the Interstate system at Kennewick on the Columbia River.
AZBuck