Earlier quoted context omitted.
While an EU citizen can at least roam in the EU fine, and a US citizen can roam the 50 states, seems that roaming outside of that would be exceedingly difficult and expensive to get appropiate visa (if they exist)
The "quiet part" you don't say out loud is you don't get work visas - you pretend you're a tourist and survive on those. However, an American could spend years just traveling the United States, and that doesn't even cover the possibility of the territories, etc.
The major checkpoint is at custom control, where choosing your words carefully will be important. I'm sure Google offers some good advice on this.