You could go to any number of countries, not just South Korea. You could go to any third world country, for example, get a good job and that would be a wonderful adventure. That's pretty much what I did and it completely changed the direction of my life.
I have a few reasons for picking South Korea, from the ease of getting around knowing only English (though I'd try to learn some Korean) to relatively higher pay than many locations that don't need specific certification in TEFL, to having a friend already there. Though admittedly I do know a bit of Spanish and have the basic grammatical/pronunciation side so just picking up more vocab would go a long way, which could argue in favor of Latin America, which also has the benefit of being closer.
Perhaps I am biased, but the Korean girls are nicer. For that fact, the boys probably are as well if I am to believe my female coworkers so you are covered on both sides.
I wouldn't look at it as leading to any type of career as much as it will be an experience that may lead to so much more. You could start of teaching English but learn Korean and get other type of work there. Most likely, you won't go in a career of teaching but it most likely will lead to some other life changing career. One of the first guys I met when I was working from our Japan office years ago was a Canadian that moved there 14 years previously to teach English. It lasted a couple of years but by that time he had learned Japanese and learned other skills and took a job in a Japanese company as well as wife and child.
But just two last bits of advise. First, why wait? If you are not working now and could get the job now, just go. Don't stop to consider and think about it too much. I woke up one morning without a thought of trying to move overseas one day until I got to work and decided I didn't like the direction they were pushing me. Within 3 weeks I had mostly given away a lot of my stuff, sold my car and started a new job in a new country. I don't know, sometimes the biggest decisions should just do and not contemplate.
Second, learn to drink. I know some good drinkers (I tend to work with a lot of sales people from many countries including Australia) but most have been shamed when they first visit there and are taken out for dinner and drinks.
I am still hoping to visit one day, I think it is the only place that I support though not visited yet for work.