Have you considered China or Taiwan? Korea requires you to go through their EPIC program to get a teaching job, I believe. But having taught in Taiwan now for two years, I can assure you the process is much less complicated, a lot of high paying jobs, and it's a beautiful place to live. A lot nicer than S. Korea, in my opinion, for what it's worth!
I was in the same position as you 7 years ago. I was just finishing up my BA and had an itch to travel. Since then I've taught in France for 2 years, Japan for three and now in Taiwan for my second year. In between, I've traveled to over 25 countries and taught temporarily in various others while passing through. I am very familiar with what English schools want in numerous countries around the world and a lot of people have told me this and that about how to get a teaching job...
The truth is your BA is the most vital. If it's a BA in Education or any teaching related field, then you're set. But if it's not, then more often than not schools will ask you for a 120-hour TEFL certificate. Getting a certificate is pretty easy, but choosing the right program to go with can be a bit complicated, as there a few out there who claim to be what they're not and give false promises, or turn out not to be very beneficial.
Before you make a final decision, you have to think about what your reason is for taking a TEFL course. Speaking from experience now, I can assure you that teaching English as a foreign language is not an easy job to step into without brushing up on your language skills knowledge and learning a bit about different effective teaching approaches and strategies to use in a classroom full of students who don't quite speak your language yet (but don't worry, you won't need to speak the native language to get a job, since most ESL schools have an English-only policy!). A good TEFL course can give you the essential tools you need to be prepared to start teaching English and not just the piece of paper saying you'reTEFL certified.
I can't tell you which TEFL course is the best because I only took one, but I can assure you it was definitely beneficial in, not only getting me a job, but getting a better salary after proving that I knew what I was doing as a teacher and not just winging it. The course I took was with MyTEFL.net. Since schools usually ask for 120hours of teacher training, I chose their 120-hour online course. It was fairly easy to finish on my weekends before deciding to eventually quit my day job and move overseas. At the end of it, they were the ones who got me my first job.
Like I said, you need to be careful since there are a lot of horror stories about a lot of TEFL providers, but I can recommend this one with confidence. Either way, definitely get yourself prepared before you head into being an English teacher, you won't regret it and you can even get lined up with a job if you go the way I went, who knows! Best of luck to you and safe travels! :)