Traveling around Seoul City is quite easy. Although it is difficult to communicate with the local, but you can find most of the traveling information for free. All you need is a tourist map or get information from tourist information center and do some homework, I believe you will be enjoying the trip well.
Gyeongbokgung Palace, Admiral Yi Sunshin and King Sejong Statute and Cheonggyecheon Stream tourist spots are all nearby, you can actually plan a visit of these places at the same day. Whereas, Insadong and Bukchon Hanok Village are around the same area.
The recommended places you should see:
1. Gyeongbokgung Palace 景福宫
Gyeongbokgung Palace was the main palace for Joseon Dynasty 朝鲜时代 and it was built in 1395. But it was destroyed by fire during the Japanese invasion in 1592 and then it was reconstructed in year 1867. The name of Gyeongbokgung was name for the new dynasty (Joseon) to be greatly blessed and prosperous.
2. Admiral Yi Sunshin and King Sejong Statute (Gwanghwamun Square)
Statute of King Sejong and Admiral Yi Sunshin was placed in Gwanghwamum Square, the commander who protected Korea during the Imjin War. King Sejong is the fourth king of Joseon which is deeply loved by Korean people. There is an exhibition hall underground with entrance at the back of King Sejong Statute.
3. Cheonggyecheon Stream
The Cheonggyecheon Stream is a stream running through central Seoul from west to east. It is a must visit place because it represented a human-centered, ecological-friendly urban area after the restore of the stream.
4. Ssamziegil, Insadong
A place that fostering the artistic and creative expression in Korean people - Ssamziegil Mall. It is situated on Insadong street which is famous for Korean traditional culture and crafts. Ssamziegil was built with the aims to support and encourage young and talented local artists.
5. Bukchon Hanok Village
After spending your time at Insadong street, you can actually paid a visit to Bukchon Hanok Village (nearby Anguk Station). You can take a walk around the alley of Bukchon Village to experience and discovery the historical culture of Hanok. There are 8 Bukchon views which you can take a nice photo.
6. Seoul Tower and Namsan Park
236.7m high of Seoul Tower offers a 360 degree panoramic view of the Seoul city, located at the top of the Namsan Park. It is the Seoul's landmark which you shouldn't miss it.