I recommend W (Two Worlds) because the lead male is a character from a webcomic. (I also started with anime and manga so I understand how the lead female feels basically meeting a manga character.)
I also second Kill Me Heal Me it had me hooked from episode 1!

Recommended and Popular Drama:

Signal (2016) - People seem to love this one and I think you will too. There's that Butterfly Effect time fantasy revolving crime-detective of cold cases. Several of these cases were inspired by true cases.

I Hear Your Voice (2013) - The guy can hear people's mind and the heroine work in the law. They kind of work together to serve justice.

Bridal Mask (2012) - Set in the 1930s when Korea was under Japanese occupation. Politic issues - Korea trying to regain independence. It was a comic, so maybe this is closer to your preferences. 

White Christmas (2011) - I didn't watch this one, but there's a group of people who really like this drama for its thriller and psychological themes. 

Check out White Christmas, it's short, memoriable and the ending will blow your mind. Psychological thriller btw XP

Welcome to the Addictive world of Kdrama

I love all KDramas but the ones with a strong story, old style or Fantasy story make me go crazy.

Gu Family Book; Kang Chi, a half-human and half-mythical being, must come to terms with his unique identity.

Empress Ki; Depicts the loves and battles of Empress Ki who exercises her influence in the Yuan Dynasty. Empress Ki is from Goryeo.


 MinaVIP:

Check out White Christmas, it's short, memoriable and the ending will blow your mind. Psychological thriller btw XP 

i can't seem to find it on here, did they change its name?

Welcome to KDrama Land, Jessica! 

You started in the right place to get recs. The forums are full of really helpful fans.

Another place where I like to look for things to watch:

https://mydramalist.com/shows/top

You can sort by country, etc on the left side which is really useful:

Top Shows Sorted for South Korea

After you get more comfortable with what you like,  check out some MDL users lists. Many have Favorite Drama lists that are super helpful; especially, if you find they have an affinity for the same dramas/actors/writers/directors/genres/etc that you you do.

MDL also has some articles that are helpful about what's out there to watch and what's really good.  Combining information you glean from different places and people about kdramas will help you save a lot time. You won't waste it on watching something you end up dropping.

All that said, I agree with a lot of other users about Healer. It's one of my favorite kdramas. Whatever you end up watching you, I hope you heart it!

Historicals

  • Empress Ki
  • Scarlet Heart Ryeo
  • Saimdang Light's diary
  • Sound of the Desert (chinese)
  • Bu Bu Jing Qing (chinese)
  • Princess Weiyoung (chinese but amazing)
  • Empress of China (one of the best dramas)
  • Jumong
  • Dong Yi
  • Jewel In the Palace
  • Jang Ok Jang
  • Ja Myung Go
  • Illjimae
  • Tree with deep roots
  • Queen In Hyun's man
  • Faith
  • Queen Seon Doek
  • Frozen Flower (Movie)
  • The flower in prision

Revenge

  • Yellow Boots
  • Flames of Desire
  • Glass Mask
  • The promise
  • Bridal Mask
  • Temptation of an Angel
  • Temptation of a Wife
  • A bird that doesn't sing
  • Queen of the Game
  • Monster

Modern (Romance)

  • Cinderella and Four Knights
  • My love Eun Dong
  • Secret Love affair
  • Noble my Love
  • Cheese in the Trap
  • Bride of the Century
  • Beautiful Mind
  • High End crush (hilarious)
  • Two Worlds
  • Descendants of the sun
  • Kill Me heal me (highly highly highly recommend as your first one)

Romance but very very sad

  • Sad love story
  • Tree of heaven
  • 1 litre of tears
  • Crying out love in the center of the world (japanese)
  • my rosy life
  • Marriage contract
  • Pained (movie)
  • Moment to remember (movie)
  • Uncontrollably fond (sad but sooooooooooooooooo good)
  • Endless Love
  • Snow Queen

Some of my favourites that I think you would enjoy! ^-^

Save me ( 4 young men, who don't have jobs, face a woman in an alley. The woman says "save me." The woman is involved with a cult religious group)

Signal ( After finding an old walkie-talkie, Detective Park Hae Yeong manages to connect with Detective Lee Jae Han in the past. Together, with the help of Team Leader Cha So Hyun, they manage to solve South Korea's most famous cold cases and reveal police corruption. )

I hear your voice ( Jang Hye Sung, a bold, sassy, thick-faced, comical, and a materialistic public defender who becomes a lawyer after overcoming poverty and painful memories from her childhood, comes to realization about society and justice after meeting Park Soo Ha, a 19-year-old boy who reads other people’s thoughts and Cha Kwan Woo, an innocent lawyer who lives a disciplined life. Jang Hye Sung is actually Soo Ha's first love after she gave a decisivetestimony on his father's murder case 10 years ago and Soo Ha promised to protect her from the killer's threats. Meanwhile, Cha Kwan Woo is Jang Hye Sung's fellow lawyer, a cheerful and an idealistic former cop even though his character is a bit slow, he is nice and pleasant with a firm idea of his principles and justice. Together they will team up to find the justice in the courtroom and solve the toughest cases with less than 1% chance of winning. )

Scarlet Heart Ryeo (When a total eclipse of the sun takes place, a 21st century woman, Go Ha Jin, is transported back in time to the Goryeo Dynasty of Korea. She wakes up in the body of 16-year-old Hae Soo, an aristocrat living with Lady Hae, her cousin who happens to be married to one of the sons of King Taejo.She soon befriends several of the princes, who find her carefree and exuberant attitude refreshing. Although she knows she should not get involved in palace intrigues over the succession to the throne, she inadvertently becomes a pawn in the struggle, as several of the Princes fall in love with her.  )