Hi Margherite! Welcome to ESL.
Mostly teaching English internationally is pretty basic stuff, and can be quite hit and miss when it comes to a quality organization or place to work at. The blunt truth about it is that almost all employers and students want white faces to teach them English, that they wouldn't get "Real" English from anybody else. Thats a hard cultural norm that you'll have to cope with, especially in South Korea.
Another difficulty you may bump into is that most employers are looking for teachers that will teach very generic English. Basically there are only two types of Ebglish calsses taught: conversational and business; and these are usually taught in such a way that there is often no difference between the two. An employer may tell you they want to get some specialty English calsses going, and have been looking for a teacher, but it never happens.

The way you want to get started is to get some good TEFL training. Universities in large cities occasionally offer this. St Giles has a one month course that is TEFL specific, but is quite expensive, and I've seen the course work, and its all very generic, and they don't seem to teach trainees ways of being creative. The Boland School in the Czech Republic and China is a five week course, but is busy. There they have 6-8 hours of training each day, then practicums in the evenings. But its affordable.

Having a TEFL training certificate greatly improves your emloyment oppurtunities, and also gives you some great insights needed when trying to teach an entirely different population than what you are accustomed to. But you need to realize that once you finish with your training, probably 2/3 of the jobs will be for teaching children, and you need to ask yourself, "Do I really love children's behavior?"

Very few of us are saving the world by being TEFL teachers, but we do have the opputunity to help improve the lives and careers of a few people.

Good luck!