I am planning (the closer to departure stage) on going to Korea to teach English. My question is general and may be easy for you to answer. Martial arts in Korea-- which fighting style is most studied, are there experienced and wise instructors, if I began lessons, would I be around mostly children? Thanks for your help.
I assume you're very aware that Taekwondo is Korea's national sport, but you might not have known that is also boasts having the most practitioners around the world. Having taught kids for a couple years, I know that it is popular among that demographic and, in the past, most Korean men who completed their obligatory military service had to obtain a black belt while serving.
Former K-Blogger Amanda from Amanda Takes Off was really into the sport and I bet it wouldn't hurt to snoop around her site. From what I understand, she learned quite a bit of Korean from her classes there.
There are other Korean and world martial arts here as well and I'm 100% certain that all of them have classes for adults with skilled instructors. As much as I wanted to imagine someone fighting kids like Kramer did (video here), I don't think you'll have many problems.

The other night, I was out having a drink with a co-worker and there was a very raucous table of Malaysians and Jordanians next to us. We joined in on the fun and found out that they were all Taekwondo referees and that they travel to Korea several times a year for tournaments and competitions. The Jordanian woman (whose name escapes me) at the table, although in her 40s now, was somewhat of a Taekwondo legend who became the first woman in Jordan to participate in the sport and is now the leading Taekwondo spokesperson for the Middle East. The point is that Korea is a martial arts destination.
Since I'm not sure where you'll be living in Korea, there's no need to suggest any dojangs just yet. Enjoy!