There's a great mix of history and modernism in Seoul. Take it all in.
If you want to get out of the city, then hop on the metro and head to the mountains. Plenty of great opportunities for hiking just a short ride outside the city.
Take a cooking class and learn to cook such Korean staples as kimchi bulgogi.
Learn taekwondo, Korea's most popular form of martial arts. If you're not that adventurous, then be sure to take in a match.
If you're looking to stay a while and make a bit of money, then look into teaching ESL. Seoul (and all of South Korea) is one of the most popular places to teach English and actually make some good money doing it.
If you're a drinker, you've come to the right place. The Koreans can drink with the best of them.