Be careful in big American cities. Most parts of the cities are safe, but all big American cities have bad parts where crime is terrible and criminals target foreigners, especially at night. I hope you will be with someone you know who is familiar with the city you are in so they can help you stay in the good parts.
But do come to America. If you are careful and watch your surroundings you will have a great time. If you have time, also go to Sedona, Arizona. It's a beautiful little town in a canyon.
A fun way to see the areas you asked about is by Amtrak trains. I've been on them and loved it.
Here's a link to Amtrak:
http://www.amtrak.com/servlet/ContentSer…
There's so much in America to see you might want to stay.
If you ever come to the Blue Ridge Mountains in Virginia I'll show you around myself!