Revenge (2005) was my first. Still one of my favourites to this day.
Playful Kiss was my first... I transitioned from anime (Ouran High School Host Club) to manga (Itazura Na Kiss) to drama... My second was Boys Over Flowers I believe.
Depends on how far fetched the question is. The first ever asian drama I watched was Skip Beat. And because of this I found other websites with those. So, after that it was Playful Kiss and My Princess. And then I couldn't stop anymore xD
Mine was probably Goong. When I first watched it I knew nothing of korean culture ( save for a few kpop songs...I wasn't a kpop fan back then ) so the palace and plot really intrigued me. I actually didn't finish that drama, I believe the first kdrama I finished was Secret Garden. It's still one of my faves.
It was Autumn's Tale - that was the first time I fell for Song Seung Hoon & Won Bin, and K-Dramas and there was no going back....
Full House (2004) was my first!
ohhhh man. boys over flowers. LEE MIN-HO.
i was like, 3 years late to watching it, it was so popular when i was around thirteen but i never caught on till ages later.
My first korean drama was Full House.. but the one that got me to dive into the place of no return was Goong!
I think my first drama was Liar Game, but Athena: Goddess of War was the one that made me realise that i love kdrama. 
Mine was Hi! School Love on, my friend introduced me to it because of Sungyeol & WooHyun and I got addicted 
Boys Over Flowers, sadly..
first k-drama was boys before flowers. not as good as he original though lol