That depends. Are you more of a city person or a country/outdoorsy type?
Typically the majority of people who come from Europe venture to the big two: Los Angeles and New York City. I cannot personally vouch for how much fun an individual will have in either, but I can speak from experience that LA is a fun city. Just MAKE SURE you have the money to visit. I wouldn't recommend anyplace like Las Vegas (overrated, not worth the expenses to travel there), Phoenix (dirty, crime ridden city), or any other location besides LA in Southern California.
If on the other hand you are more of an outdoorsy type...I can recommend the following places: Oregon, Washington State, Northern California, Colorado, North Carolina, Vermont, and Southern Florida. Texas is even a fascinating state. Don't listen to all the rambling bullshit you hear others say about the South. It's not as bad as people make it out to be, they are just nitpicky political arguments. If you do go to Texas check out San Antonio or Houston. They are two of the most sprawling, beautiful cities to visit.