Surgeon Bong Dal-Hee is definitely the one to start. Very popular show, it aired against Dal Ja's Spring, and consistently outperformed it. It's a bit like Gray's Anatomy (and in fact, it was originally derided as a GA clone), but in some ways (mostly due to consistent serialization and Korean knack for soapy plots) , it improves on the American model. The action comes hard and fast; by the end of episode 1, our plucky heroine's patient has already died!
It's really well done, you got loads of crises, romantic bits, very well-done hospital politics, board examinations, and some poo poo humor. Gotta have poo poo humor when you're doing medical drama!
Off note with Bong Dal-Hee is the lead actor Kim Bum Soo, who's a well known actor in Korean movie comedies. This was his first real TV lead, and one of the few times he's ever had a true non-comedic role. And he's great. Walking like a linebacker and shouting from a tightly wound, cruel mouth, it's a joy to watch him chews whole scenes out and make everybody feel THIS small. Also interesting is that the two support roles are played by Kim Min Joon (of Damo and Ireland fame) and Oh Yoon Ah (the Kim Hye-Soo of our generation?), as a divorced couple. They played opposite each other in a previous standout drama Someday (starring Bae Doo-Na and produced by the same company who made Alone in Love.) They carry their big chemistry here.
If you're looking for a serious hospital drama, then check out The White Tower, which is considered one of the best dramas of the year. This is an adaptation of the Shiroi Kyoto (White Tower) dramas, which itself has classic status. Both White Tower and Surgeon Bong Dal-Hee ran roughly during the same period, and a lot of people were comparing the two. BDH has a lot of soapy fun elements, but WT is a serious drama involving a power politics. It also stars Kim Myung Min (Bad Family), who's up for actor of the year for his role in White Tower. Coffee Prince fans, take note: "The Voice" Lee Sun Gyun is the 2nd lead in the show, playing friend and rival to Kim Myung Min's character.
I think I read somewhere that we would see a sequel to General Hospital (MBC hospital show from 1994) in 2008, but I doubt that happens if they do New Heart. And I really wanted to see who'd they get to play Sonny and Carly!