Seoul is a huge and crazy city. If you're anything like me you probably landed and immediately thought you have not enough time. I would compare Seoul to NYC. Tons to do, very cultured and international and a GREAT subway system (full guide to using the subway here). So where do you start? Here are the top things to do in Seoul and some things that CANNOT be missed.
Visit the Bukchon Hanok Village and specifically Simsimheon. For around $12 you can get a tour of a refurbished Hanok and enjoy quiet time in the garden with the included tea.
Old Seoul is gorgeous but new Seoul is just as cool. Check out DongDaemun Design Plaza which feels like a UFO that landed in the middle of Seoul.
Palaces, Palaces and more Palaces. There are 5 beautiful palaces to see in Seoul - here is a quick ranking if you only have time to hit 1 or 2.
Explore the Ihwa Mural Village. Although the best mural is now gone it still has some amazing things you should see.
Namsan Mountain (yes Seoul has a mountain right in the middle of it) has a nice path up to the top if you want to get some exercise in while visiting Seoul. (who doesn't like to exercise while on vacay) Or you can take the gondala to the top like I did.
Explore The Palaces at night. Seoul does a great job at lighting them up in all of their glory.
While wandering at night don't forget to just admire the craziness of Seoul. This isn't a specific street, every street looks like this at night.