Yeah, it's really easy. All you have to do is get to a foreign county, go to the US Embassy there, turn in your passport and tell them that you want to renounce your US citizenship. That's it.
Hard part is going to be finding a country that is going to let you immigrate. And, pretty much once you get outside of the 1st World, life gets crappy and cheep in a hurry. I take it that you're not thinking of going to Mongolia or Ethiopia or the Congo, but NO ONE who has grown up in the US can even begin to picture what day to day life is like in the Third World. Most of those countries don't even have a word for tolerance.
Unless you have special skills, it's hard to get into Europe, New Zealand, Canada or Australia. And forget Japan, if you're not Japanese they make no secret about not wanting you. You can go there to work, and they like that, but citizenship is all but impossible. You can get a permanent resident status after 10 years or so, but that's not the same thing.
Among the hardest places are Germany, Israel if you're not Jewish, Switzerland (12 years and then you have to go in front of a committee that has the power to deny you citizenship if they don't like you.)
OK, first, I'm going to assume you are white. If you are not then this is a whole different deal. And if you don't have at least a BA from a regular school (as opposed to some bible college) forget it.
Join their army. It's the time honored way.
Marry someone in that country.
Do you have specific skills that are needed in that country?
Even better, a job - this is the BEST way to go, that way the company does the immigration stuff for you, but you have to have real mad prop skills at something they really need before they are going to pay you to move halfway around the world and go through all that.
Do you have enough money to support yourself without working? - lots of places are going to insist that you have X amount of money.
Almost all countries have both health and character qualifications on top of that. And by health, I mean the most invasive physical exam you're ever going to get by a doctor of their (not your) choosing. I would think that just about 80-90% of the 120 or so different nations would exclude trans persons just on that basis alone. So your left with something between 10 to 20 realistic choices.
And ignoring UN declarations is pretty much a world-wide hobby.