I'll also add the Insadong area as a great place to base yourself. The area itself has a vibrant shopping/street food/restaurant scene. It's also within easy walking distance of Gyeongbokgung Palace, Bukchon Hanok Village, and Changgyeonggung Palace & Secret Garden. If you can walk 30-40 minutes, you can also reach Namdaemun, Myongdong, Gwangjang, and Dongdaemun. Hongdae (which was a great area to visit) and Gangnam (didn't have time to visit, unfortunately) would require subway or taxi. Of all of the neighborhoods we visited, I found Myeongdong to be the most overrated and Gwangjang to be the most awesome.