Gangnam became one of the most well known places in Korea now because of the song, Gangnam Style. Gangnam(around Gangnam subway station) is popular not only among young people but also many business men. Samsung HQ is located there and many companies are located around Gangnam.
Among trendy young Koreans Hongdae(around Hong Ik university) is the one you must go. Many trendy cafes, restaurants and pubs are really popular among young tourists as well. For me, I love shoe shops in Hongdae. They sell trendy, good quality shoes in affordable prices.
Where could be called as Hongdae in Jeju? I can say Lee Jung Seob street should definitely be the one.
Lee Jung Seob street is located in the heart of Seogwipo city(along Olle route 6). Lee Jung Seob is one of the most beloved oil painters of all Koreans.
Lee Jung Seob st. is next to Lee Jung Seob gallery and a humble cottage his family used to live.
The gallery entrance fee is as humble as his cottage so enjoy the famous oil paintings and take a slow walk along the street. Fancy cafes and small galleries are good backgrounds for your photos.
Below URL has a few recent photos of Lee Jung Seob street.