It's a good thing that you're young. Countries are more likely to want to keep you around if they know you're healthy and able to work. On the other side of the coin, moving as a means to take advantage of resources is probably not ideal from their perspective. As an American, there won't be a lot of other places exactly rooting for you to come and stay either. For example, an old Canadian friend of mine just moved to Australia because all of the common wealths have an under 30 rule that grants them a year visa at another country for practically nothing. After that year, it's been a heck of a lot easier for her to stay. You on the other hand won't be getting any handouts. Most of the world still thinks that all Americans are rich so getting through the bureaucracy will cost more for you. Speaking of, I'm in the final stages of gaining permanent residency in Canada at the moment. Thing is, I had to prove that my job could not be done by a Canadian. I did this through a series of tests and a butt load of paperwork. It also really helped that I went to grad school in British Columbia. I don't know what kind of a chef you are, but if you don't have any special pedigree or certificate or something, it might be more difficult to stay - having something quantifiable is key.

From what little I know of your situation, I can think of two ways to get settled some where else:

1. Give up your citizenship and apply as a landed immigrant. 2. Move alone first, without the family, settle into a reliable job, get onto a path of residency, and then bring the family out.

Hope that didn't scare you.