(Edit: Wait, is there an actual pathway to permanent residency/citizenship this way, or is it only a temporary work visa?)
Until that point you'll usually have to do something like check in with immigration once a year, dependent upon what visa (work, education, tourist, etc) that you end up on. But like another poster said, you don't really need to worry about most of this to begin with, as at most places you'll be granted a visa on arrival that'll be good for more than enough time to find a job. The most you might want to do is drop by the embassy of wherever you plant to go to. They can give you more information, and also set you up with a visa that will generally last much longer than the visa-on-arrival if desired.
But the main thing I'd say to anyone considering moving countries is that most countries aren't highly desired immigration destinations like the US or Western Europe, and thus most countries don't have strict and tightly-controlled immigration systems like Western countries do. If you want to go spend a few years in Thailand as a Westerner, it's laughably easy - just buy a flight and you'll probably figure it out once you're there, as your Western dollars/pounds/euros will go a looooong way.
(I apologize to any Thai people reading this; I know that digital nomads and other wealthy foreigners aren't universally appreciated in the countries they park up in. Am just sharing my own experience.)
I wouldn't want to live in that part of the world for the rest of my life, but the time I did spend travelling was the best thing I ever did and I have no regrets. I wrote about it here if anyone's interested:
Depending on where you go to (and what type of institution) a bachelor's alone is sufficient. The more exclusive places might require a teaching certificate, but those are easy to get and, in any case, such requirements tend to be quickly relaxed when talking about harder to fill positions, like calculus teachers. You might find you enjoy this. If you don't, it can give you the foundation to find something else you might enjoy. Many companies are also constantly looking for skilled native workers. I've friends working in everything from web design to rehab counciling.
I'm not very happy with the current state of the US to say the least, but being born American and simply natively speaking this language is just an unbelievably valuable gift that one should never take for granted. It's really a gateway to the world, even if it's one hell of a scary gateway to step through the first time.