I think part of the reason for her popularity is because she was in so many good movies: Barking Dogs Never Bite (1999), Take Care of My Cat (2001), Sympathy for Mr. Vengeance (2002), Linda Linda Linda (2005), The Host (2006), Air Doll (2009), A Girl at My Door (2014). I can't think of any other Korean actress who was in as many good movies as her.
I suggest starting with her earlier movies because those are the movies that made her famous. I watched them a long time ago, so I don't really remember which one had the most emotion.