I studied a semester abroad in Florence, Italy (Firenze) and it was awesome. Personally I was invested in the trip as I have Italian ancestry and a specific interest in art and music, but even beyond that spending time immersed in another culture is an amazing learning experience and you can only get it by going. Especially when money isn't an issue (that's often a pretty big hurdle for people), absolutely go.
Another thing to consider is that doing things like this gets much, much more difficult after college, if possible at all. Paying for these kinds of trips can be an issue when scholarships aren't available, and rarely (often not at all) will you be able to just leave your job for 1 or 2 months to be able to travel and immerse yourself in a particular country and culture. Usually you're lucky if you get a week.
Also, in regards to South Korea as a location, I have a close friend who spent a summer there in college. He absolutely loved it. As a culture, the people were very polite and courteous, which is generally agreeable to us midwesterners. Plus the country is very developed, they have a strong economy, and there's lots to do.
This same friend actually loved it so much that after a year of working, he decided to quit his job and moved to South Korea for a year or so and taught high-school level English. (The country has a special program to bring in native English speakers to teach)