Seoul has several locations that are popular for shopping and also experiencing the culture.
Markets
Myeongdong
This famous bustling shopping area is north of Namsan and includes many familiar stores like H&M, Forever 21, Krispy Kreme Donuts and a lot of interesting street food. This is a great place to try Dak Galbi - a spicy chicken meal. It is also known for its plethora of cosmetics shops.
How to get there: Myeongdong Station (subway line 4, exit 6 or 7; bus 421 from Itaewon)
Namdaemun
Namdaemun is an old shopping area in Seoul not far from Seoul Station. There you will find various stores to browse, from coffee shops, clothing, Korean restaurants, and Alpha, which is a great place to get school supplies. Just outside of the market is a point of interest: The Namdaemun Gate.
How to get there: Hoehyeon Station (subway line 4, exit 5; bus 405 from Itaewon). The market will be to your right.
Insadong
Insadong is a popular market area with Korean souvenirs, traditional tea houses, ceramics, and artwork. Just outside of Insadong is Pagoda Park, where a relic pagoda stands encased in glass.
How to get there: Anguk Station (subway line 3, exit 6)
Dongdaemun
Dongdaemun is located in the northeastern section of Seoul and is a popular shopping area if you want to get the authentic Korean market experience. Here you will find shoes, accessories, fashion, souvenirs, food, and various stores selling inexpensive items. There is the Kwangjang Market in Dongdaemun, which is a large shopping section within the alleyways of the local buildings. Other points of interest there are the Dongdaemun Design Center, Dongdaemun Gate, and the Seoul Wall Museum.
How to get there: Dongdaemun History and Culture Park Station (subway line 2, 4, 5, exit 14) or Dongdaemun Station (subway line 1, 4, exit 7 or 8; bus 420 from Hannam-dong)
Noryangjin Wholesale Fish Market
Noryangjin has an assortment of cheap street food and coffee shops that cater to the poor college-age population there. If you cross over the bridge, you can explore the popular fish market.
Malls
IPark Mall
This building houses a variety of stores such as eMart in the lower levels, CGV theater on the 6th floor (how to buy a ticket here), and in between are various restaurants and shops like Modern House. Modern House has many Western-style home furnishings. IPark is connected to Yongsan Station and is right by the Yongsan Electronics Market.
How to get there: Yongsan Station (subway line 1; bus 400 from Itaewon)
New Times Square Mall
New Times Square Mall is a modern shopping center complete with what you would expect in a large mall.
How to get there:
1. Yeongdeungpo Station (subway line 1). Leave the station through Lotte Department Store. Cross the street, and continue for 200 m. Turn left, and walk 150 m forward to arrive at Times Square.
2. Yeongdeungpo Market Station (subway line 5, exit 4). Walk for 10 min.
3. Mullae Station (subway line 2, exit 4). Walk for 7 min.
COEX
COEX holds COnvention centers, EXhibition halls, and "the largest underground mall in the world." Inside is a high-quality aquarium that is fun for all ages. There is also an interesting kimchi museum if you were ever interested in its history and various forms. Outside COEX is the distinctive World Trade Center building and a G20 Summit memorial.
How to get there: Samseong Station (subway line 2, exit 5 or 6)