As an Ameriacan as well I must concur. Do not miss the parks (including Yosemite which is in California and not yet mentioned)
Yes a car is reccomended. This country is huge, with the"good parts" located far from one another. I agree with thearlier comment regarding New orleans ( I lived near there at one point, if you can put it on you agenda.) And do not miss NYC. It really cannot be put into words.
If yu have specific questions I 'm more than willing to oofer suggestions!