Are you sure you are ready for kdrama lore yet? :)
I also started with kmovies (about 10-12 years ago), there are a lot of good ones. But one day I asked my friend to buy some Kmovies on his holiday abroad and he accidentally bought a drama with same title as movie. That was my first step into Kdrama. And now - My name is "Beep, beep" and i am an addict.
Some of my suggestions (I will cut out the ones from previous posts - they are quiet good "reco"):
Ghost (2012) (crime, hackers, detectives, conspiracy, "phantom of the opera" ...)
History of the Salaryman (2012) (business comedy, "Romance of the Three Kingdoms" parody, with some drops of "melo", in my opinion the best Kcomedy)
IRIS (2009) (unglamorous spy drama, betrayal, revenge,
listening IRIS OST - track "Destiny Love" + memories of drama = my heart stops beating)
The King of Dramas (2012) (comedy/parody of drama making, fast story + of course what would be kdrama without melodramatic moments)
All In (2003) (old style kdrama - could be inconvenient, gambling, betrayal, revenge)
Bad Guy (2010) (suspenseful revenge melodrama, I didn't think I would watch this but I got immersed with story, acting and soundtrack)
H.I.T (2007) (detectives, serial killers, for true crime fans maybe,...)
Tazza (2008) (another gambling revenge drama, the recipe "try hard -> get betrayed -> go to prison -> get better with enemies of your enemy-> take revenge" is quite common in kdrama and kmovies :)
Sign (2011) (crime, investigation, forensics, speedy story,...)
ETC.
If you manage to see and like any of this drama, just ask for reco or check my dramalist.