Bleak Night and Pluto
Hello,

For me the best korean director is Im Kwon Taek, but he makes art movies which are a little slow and contemplative. You can try :
http://mydramalist.com/1741-chunhyang
It's a pansori very famous in korea.
I hope you will like it ;)

3-Iron
Train to Busan
Tunnel
Veteran
The Throne
The Chaser
The Himalayas
The Handmaiden
Way Back Home
Miracle in Cell No. 7
Windstruck 
Hello Ghost
Shoot my Heart
Tower
Cold eyes
I really really liked Always (romance and melodrama, but beautifully shot) and My P.S. Partner (steamy romance, explicit sex scenes, very different from the usual fluffy rom-coms).
If you feel like crying: Hope
My sassy girl (comedy))
My ps partner( ji sung , comedy, romance)
A werewolf boy(hearttouching)
My tutor friend 
I saw the devil( thriller)
Fabricated city highly recommend....i even ask my brother to watch who never really like korean movies and he said it was really good...????????
A Werewolf Boy
Coin Locker Girl
My Annoying Brother
The Man From Nowhere
Always