Seoul is very, very spread out, and the easiest and cheapest way to get around is the subway. Fortunately, most of the tourist attractions are accessible by the subway, and the subway is sparkling clean. You can get an English maps, and all of the signs are in English. Station stops are announced in English on all of the subway cars. Never attempt to drive around Seoul, as you will see gridlock and impatient drivers everywhere. Don't take a taxi unless if you are ready to explain where you want to go in the Korean language. Don't try to use the city buses, the network is simply too convoluted, and the drivers can drive like total maniacs.
If you want to enjoy the traditional side of Seoul:
Visit one of the many temples located throughout the city, including Gyeongbokgung (Subway Line 3), Changgyeonggung and Changdukgung. You can also visit the folk village located in the Southern part of the city, right on the outskirts. Also visit Insa-dong, which is the art district of Seoul. It's located not far from Gyeongbokgung temple.
If you want to explore the Japanese colonial era and Korean-war era history of Seoul:
Visit the Seoul Museum of History near Gwanghwamun station on the 5 subway line, and IIRC there's an outdoor exhibit near Yongsan (near Ichon station) that has various WWII-era military planes.
If you are into the clubbing scene:
Hang around near the vicinity of Hongdae, which is accessible on the 2 subway line at Hongik University station.
If you would like to take a look at the daily life of a contemporary Korean, or just explore the cityscape and modern side of Seoul:
Visit the COEX mall located at the Samseong subway station on Line 2. Go shopping or just hang around Myeong-dong, or dine. Walk around City Hall and walk along Cheonggyecheon, an urban stream that was recently recovered and rehabilitated as an attraction, from something that was covered over by a highway.
Visit Gangnam, which has a mood similar to the Upper East Side area of Manhattan; full of affluent, conservative people that live in highrises adjacent to various fashion boutiques and luxury car dealerships. If you see lots of non-Korean cars on the street, it means that you are in a wealthy area, as owning a foreign-made car is a symbol of prestige for Koreans.
Visit the Seoul Tower, located at the summit of Mt. Namsan, reachable by cable car. This offers you a spectacular view of the sprawl of Seoul.
The DMZ is only around 50kms away from Seoul, and a day trip can be easily arranged by a tour company at the airport, or at a tourist information center.
There's a tourist trap called Itaewon, full of merchants that will try to rip you off. However, it also happens to be the most diverse part of Korea, where you will only see a handful of Koreans among the huge crowd of tourists from all over the world.
If you're going now, prepare to dress up for the colder weather. Hope that you enjoy your stay!
The biggest mistake made by most human beings: Listening to only half, understanding just a quarter and telling double.