I am pretty confident that this was the first ever K-pop song I heard:
10 Minutes by Lee Hyori. The song was released in 2003. I don't remember exactly when I first heard it, but 2004 and 2005 are my best guesses. Back then I actually thought the song was Japanese. I liked this song a lot, but I didn't get more into K-pop back then and even went to forget about this song.
Later on when I started to get into K-pop music few years ago (2011 maybe), one of the first artists I came across was Lee Hyori (since she is so famous). The name sounded familiar and then I remembered the song I had heard years before. Of course I went to look for more songs from her. I really liked her song called U-Go-Girl and it is still one of my favourite songs of her. From my experiences K-pop music from when this song was released tend to be more silly than nowdays. Even though I love more recent K-pop music too, these sillier songs from few year back are a lot of fun. And they are very catchy!
For me getting into K-pop happened through two different paths at the same time. Firstly through J-pop. I think this route comes quite naturally for anyone whose into J-pop. A lot of K-pop artists also make songs in Japanese (for bigger market).
Like in my J-pop post I mentioned, I really got into Japanese girl groups. At some point I started to come across also South Korean girl groups too (mentions, links to videos, YouTube recommendations, etc...). I wasn't purposely looking for Korean songs, but since Japanese and Korean music markets have so close connection, K-pop sneaked into my life.
But of course I was very interested to find more girl groups to listen. One of the first K-pop acts I came across was the very famous Girls' Generation, also known as SNSD. Since it was a girl group, I obviously decided to listen to them. I totally fell in love with their song Run Devil Run. It was actually the first song I added on my K-pop playlist on YouTube.
The other route to K-pop was through a friend. I suppose this is a quite common route. My friend (whose awesomely active [note the sarcasm] Twitter account is @De_Arctic_Fox) had found South Korean music recently and send me couple of links to songs she liked.
The first time she linked me a song, I believe it was this song:
Despite this blog post's title, this song is rock (or pop rock?). The band is called TRAX and the song is 오! 나의 여신님 (Oh! My Goddess, in English). At first when I heard this song I didn't like it much. Then one time I got it stuck in my head and listened to it again resulting into me liking it. Now that I listened to this song after a while, when looking for songs to this post, I actually liked this a lot. Tastes change, obviously.
One other time my friend send me this song:
Wedding Dress by Taeyang. I immediately like this song, and I still love it a lot. It is very beautiful and touching, even though I don't understand anything else than the few English words. The music and his singing are enough. The video adds up to it all.
So, these were few of the very first South Korean songs I heard, probably all within ten, or less, first ones. From these on, I started to more purposely look for K-pop songs, especially girl groups. I believe the first songs you hear from a genre/band/whatever are very important, since they most likely define whether you get interested in it or not. These at least did the trick for me. Also, the first songs are important because you kinda base on your later music on the first ones. That's at least what I think.