Here is a list of things that I believe you must do if you are ever in Korea. I really enjoyed myself and would definitely recommend other people to go.
1. Ewha's Women's University
Ewha was one of the best and cheapest shopping areas we went to. The market is right around the women's university and there are a plethora of options from bags to shoes. I would highly recommend it has a stop, you should definitely find what you are looking for in this area. Even the food was very inexpensive.
2. Hongdae
Hongdae is another well-known area for it's shopping. And at night time due to its proximity to the university, it is very lively. There are many clubs right next to each and there are mostly electronic. We went to cocoon and did not leave until 6 a.m., which is how late most clubs are open until 6 a.m.
4. Han River Cruise
We went on a 30-minute cruise during the day which it was nice but I would recommend trying one closer to night time because then the lights are lit.
5. 63 Building
6. Palaces
We went to Gyeongbokgung Palace which had two museums but there are many Palaces you can choose.
7.Hangang Park
Hangang Park should definitely be a destination on the list as many Korean Dramas have a scene at the Park. I actually ended up taking pictures at a location where My love from another Star and other dramas where filmed.