Skeptical beginner looking for something actually good (kdramas)

Healer - brilliant writing, compelling characters, awesome acting, fantastic chemistry, intriguing and mysterious storyline and a romance that is so perfect it just makes you laugh, cry and giggle every time. It has romance but it doesn't rely on that to propel the story, the character relationship, the big bad villain and the character development is very much a part of what will me you stay and tuned in

There are more dramas, but you should really start with Healer.

For sites to watch, dramafever and viki wpuld be a go to places. As well as netflix. 

Some Korean dramas from recent years that I can wholeheartedly recommend:

• Argon (journalism, only 8 episodes so that's a plus I guess)
  
• Stranger (crime; not really a procedural drama, though; it had this big narrative going on)
  
• Dear My Friends (drama/melodrama about a group of friends who had grown up and grown old with each other over the years; be prepared for the sheer loudness of its actors on earlier episodes, though)
  
• The End of the World (a 12-episode medical zombie apocalypse, which is more about the follies of people than the violence of zombies)
  
• A Tree with Deep Roots (historical drama about the making of hangul/Korean alphabets)
  

For a list of legal streaming sites, please refer to THIS THREAD

Right now i am watching Chief Kim and it is ridiculously entertaining. Very fun and the story is engaging + it has no romance.

Chicago Typewriter is also something i recommend but i feel like it is not a typical kdrama. It has romance but it was quite easy to ignore it and pretend that main leads are just friends :D. It is a mix of history and current time (but not time travelling) and has really interesting atmosphere.

If you want to give a shot to a romantic drama, you must definitely watch Reply 1997. It has brilliant writing and it is really fun. First episode is a bit overwhelming but it is worth it. 

I also cannot help myself but recommend 2 one-episode tv specials (which you can find on youtube) because they are really good:

The Legendary Lackey - high shool comedy which basically introduces you to every korean high school dramas.

The Red Teacher - this one is one of my all time favourites. Idk, its just good.

As I haven't watched dramas that were recommended by other users above I made an advance search for the highest rated Korean dramas and the list included a lot of the drama suggestions by users above.

Healer

Kill me Heal me

Hello Monster

Signal (it is a detective drama but the storyline is superb, it's very intriguing)

I'm Sorry I love you

Just Between lovers

Come and Hug me

I'm pretty picky myself, so I have a short list of completed dramas. A large part of enjoying asian drama  is understanding that they have a different style  than most  western shows.  However, that doesn't mean they can't be enjoyable. Here are a few of my absolute favorite. (Click to view.) 

** Circle - Taking place in both the present year 2017 and in the year 2037, when people believe that their future depends on humans controlling and regulating their emotions, this drama is a sci-fi mystery. When alien visitors arrive on Earth, humans learn new skills from them and take advantage of them; what ensues is a story of alliance, resistance, and love. Myster/Sci-Fi

** Just Between Lovers - Lee Gang Doo dreamed of becoming a soccer player, but an accident ended those dreams. His father died in the accident and he broke his leg. Gang Doo spent 3 years in rehab. Now, he doesn't have anything and just wastes his days. Gang Doo then meets Ha Moon Soo.Moon Soo makes architecture models, she also endured an accident that killed her younger sibling. Moreover, she has suffered from the guilt that she survived. She has lived her life earnestly while suppressing her feelings. Trauma/ Psychological

Reply 1988 - Five childhood friends, who all live in the same Ssangmundong neighborhood of Seoul, lean on each other to survive their challenging teen years and set a path for their futures. Family/ Coming of Age     

 I Can 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. Thriller/ Mystery/ Supernatural

As far as the best website to watch asian drama, I recommend two:

Kissasian

Dramacool

I just wanted to chime in that I don't like American crime, medical, legal, military, etc shows so avoided them for years. I regret it so much now because there are so many great ones. SK has more focused storylines since there is usually only one season, so they don't drag as much. The feel and focus is often way different than American counterparts. :)