Korea free and easy: Seoul Gyeongbok-gung(경복궁,景福宮),

Gyeongbok-gung.
The grandest and the most beautiful of the palace.
So if you have only time for one.
This should be the one to visit. 

Catch the guard changing ceremony.

Enjoy the beautiful scenery.

Museum

Although this not the traditional costume of Koreans, is quite fun to try this out too. and is free.


Closed
Tuesdays
Operating Hours
March-October: 09:00-18:00
November-February: 09:00-17:00

Admission Fees
Adults (ages 19-64): 3,000 won / Group (over 10): 2,400 won
Children (ages 7-18): 1,500 won / Group (over 10): 1,200 won

Interpretation Services Offered
Tours depart in front of the Information Center at Heungnyemun Gate (흥례문).
Duration: 1hr - 1hr, 30min
Tour Schedule:
English: 11:00, 13:30, 15:30
Japanese: 10:00, 12:30, 14:30
Chinese: 10:30, 13:00, 15:00

Directions
Gyeongbokgung Palace Station (Seoul Subway Line 3), Exit 5.
OR,

Gwanghwamun Station (Seoul Subway Line 5), Exit 2.
Performance Times

January 1-December 31 (Closed on Tuesdays)
Sumunjang(Royal Guard) Changing Ceremony
10:00, 13:00, 15:00 / three times a day / 20 minutes per ceremony
Gwanghwamun Gate Guard-on-Duty Performance
11:00, 14:00, 16:00 / three times a day / 10 minutes per ceremony

For more information on hours, and how to get there.

http://english.visitkorea.or.kr/enu/SI/SI_EN_3_1_1_1.jsp?cid=264337

Integrated palace pass
http://mizztoee.blogspot.sg/2013/02/korea-seoul-deoksugung-palace-and.html

If you manage to visit Korea in Autumn, you will be able to catch the beautiful foliage.