Start your day by immersing yourself in Korean culture with a visit to Gyeongbokgung Palace. It's one of the most iconic structures in Korea and was the home of the Joseon Dynasty. After that, head over to the Bukchon Hanok Village where you can see traditional Korean homes and enjoy a cup of tea. Lastly, spend some time strolling around the Insadong area where traditional goods, shops and eateries stand.
Get an amazing panoramic view of the city from the Namsan Tower, then head to Namdaemun Market, one of the biggest street markets in Korea. Lastly, check out Myeong-dong for its shopping and street food scene.
Take a leisurely walk along Garosugil, which is known for its trendy shops, cafes and restaurants. Afterwards, visit the Leeum Samsung Museum of Art to check out some of the most beautiful and innovative works of art by local artists. Wrap up the day with a relaxing stroll in Hangang River Park where you can rent bikes or take a river cruise while enjoying the beautiful scenery of the Han River.
Enjoy a fun and exciting day in Lotte World which is Seoul's version of Disneyland. Then, finish the day with some shopping at the Coex Mall, one of the largest shopping centers in Asia that also houses an aquarium and a library.
If you have more time, consider a side trip to Jeju Island or a visit to the DMZ. For those that have less time, it's okay to skip Coex Mall, as it doesn't have any unusual experience to offer. Always check weather forecast regarding smog/rain. Lastly, if you plan to visit more palaces and historical sites, it's best to purchase the integrated admission pass for all the palaces for the cost-efficiency.