A lot depends on how you are traveling, but some interesting cities would be Washington DC which houses the Smithsonian Institute as well as other wonderful free museums, New York City has endless attractions (but maybe not so different from other large cities of the world). San Francisco is a unique city and close to the Redwood forests, Napa Valley wine country and the beautiful Pacific coast. Two places that are fun if you like country music would be Nashville, Tennessee and Branson, Missouri. Both of these cities have many country music shows where you can see top names perform. Nashville also has a wonderful museum, the Country Music Hall of Fame.
If you are camping or renting an RV, then you should definitely try to visit some of this country's National Parks as they are amazing places. The Grand Canyon, Yellowstone National Park, Arches National Park, Bryce Canyon...there are so many beautiful places to see! One of the problems with visiting the US and trying to see a lot of the country is that everything is so far apart. I hope that you will have a lot of time to explore the cities as well as sample some of the natural wonders of the U.S.