re: 3 weeks in the USA.

I live in Dallas and it's a fine place to live. But there is not a lot here for tourists. FYI, Dallas is a liberal city, just like all Texas cities where you have educated people. It's the suburbs full of people afraid of blacks and Mexicans and the rural areas of Texas that give us a bad name. If I were visiting one city in Texas without a car it would be Austin, our state capitol. Beautiful area with a lot of Texas history. But you might also like Ft Worth since you can see some real Cowboys in the stock yards area. And ft. Worth has some great art museums and a nice downtown. Going with the No car theme, other cities I would suggest would be Chicago, San Francisco, Seattle and Portland, Oregon. Nashville would be another cool option.
I guess the question is where are you from and what do you want to experience?