Mad Irishman, Mount Desert and Bar Harbour was my suggestion, too. It's beautiful, and I would feel okay travelling there alone.
If you are driving, those states, Maine, Vermont and New Hampshire are small, and you could plan a great trip if you drive.
NYC is awesome, but expensive. I am sure people here, would be most helpful in advice - ie, safe means of transport (cab, subway or bus?), where to stay, etc. There are things in NYC, that make it a 'must-see'.
Loving the Carolinas, too, and W. Virginia - just stay at a Best Western or better, and talk to the locals. People are so friendly.
Las Vegas - just don't get drunk, and you will be fine. Accommodations and cheap food, great shows, and the casinos are amazing to see for the first time. It's true - there are NO clocks in casinos. People forget the time when they are gambling.
LA - Manhattan Beach, home of many surfers, and another great city. You need to drive if visiting CA.
PS. Been to all those places, I want to go to TEXAS. Lots of apostates there, and it's an incredible place, or so I've heard. xx