Here's an ideal itinerary for 4 days in Seoul:

Day 1:

• Start your day with a visit to Gyeongbokgung Palace, one of the most iconic landmarks in Seoul. You can watch the changing of the guard ceremony and explore the palace grounds.
• After that, head to Bukchon Hanok Village, a traditional Korean village with beautiful hanok houses and narrow alleys. You can also visit the Bukchon Cultural Center to learn more about the history and culture of the area.
• For lunch, try some delicious Korean street food at Gwangjang Market, one of the oldest and largest traditional markets in Seoul.
• In the afternoon, visit Namsan Tower, a popular tourist spot with stunning views of the city. You can take a cable car or hike up to the tower.
• End your day with a visit to Myeong-dong, a bustling shopping district with plenty of street food, cosmetics shops, and fashion boutiques.

Day 2:

• Start your day with a visit to Changdeokgung Palace, a UNESCO World Heritage Site known for its beautiful gardens and architecture.
• After that, head to Insadong, a neighborhood known for its traditional Korean crafts, tea houses, and art galleries.
• For lunch, try some Korean barbecue at one of the many restaurants in the area.
• In the afternoon, visit the National Museum of Korea, the largest museum in the country with a vast collection of Korean art and artifacts.
• End your day with a visit to Hongdae, a trendy neighborhood known for its street art, live music, and nightlife.

Day 3:

• Start your day with a visit to the DMZ (Demilitarized Zone), the heavily guarded border between North and South Korea. You can take a tour to learn about the history and current situation of the Korean peninsula.
• After that, head to Itaewon, a multicultural neighborhood with plenty of international restaurants and shops.
• For lunch, try some delicious Korean fried chicken at one of the many restaurants in the area.
• In the afternoon, visit the War Memorial of Korea, a museum dedicated to the Korean War and the country's military history.
• End your day with a visit to Gangnam, a trendy neighborhood known for its high-end shopping and entertainment.

Day 4:

• Start your day with a visit to the Jogyesa Temple, the center of Zen Buddhism in Korea. You can participate in a temple stay program or simply admire the beautiful architecture and gardens.
• After that, head to the Noryangjin Fish Market, a bustling seafood market where you can buy fresh seafood and have it cooked on the spot.
• For lunch, try some bibimbap, a traditional Korean dish made with rice, vegetables, and meat, at a local restaurant.
• In the afternoon, visit the Seoul Museum of Art, a contemporary art museum with a diverse collection of Korean and international art.
• End your day with a visit to the Dongdaemun Design Plaza, a futuristic building designed by Zaha Hadid that houses a variety of shops, galleries, and exhibitions.