The best and most cost-efficient means of travelling in Seoul is by subway. Techies and gadget geeks can visit Yongsan where you exit the station straight into the bus terminal and next to the Tech Market. Those looking for culture and history should go to Jongno where you can visit Gyeongbokgung Palace or walk down Insadong for handcrafted goods. The Namsan Tower visible from most parts of Seoul (unless you're surrounded by skyscrapers) can be reached by cable or by walking. Across from the tower is Myeongdong, an affordable alternative to the expensive malls, with floors upon floors of shopping at Migliore and plenty of shopping on the streets as well.
Amusement parks are visited, not only by youngsters, but by older visitors. Lotte World tends to be popular. If you want something more relaxing, at least to locals, try visiting a jimjilbang, which can cut costs for accommodation and provide a unique sauna experience.
Hongdae tends to be popular on Sundays for local art and a cup of coffee, and on weekend nights, turns into one of the spotlights for clubs, bars and restaurants. It is one of the more diverse cities besides Itaewon, which tends to be an expat hub for those missing the international scene. Young professionals and business travellers are encouraged to visit Gangnam where locals will grab a night out on the town still dressed in their business suits. Definitely try Korean BBQ and kimchi (preferably with BBQ and rice if it is your first time) for the full cultural experience. If you're more daring, visit one of the unlimited number of nohraebangs (karaoke bars).