I haven't watch those dramas either.
I agree that watching korean dramas help to understand korean culture.
I recommend these Dramas: These Dramas were really popular and are worth watching.
-Winter Sonata (겨울연가)
-Beautiful Days (아름다운 날들)
-Goong (궁) a.k.a Princess Hours
-Full House (풀하우스)
-All In (올인)
-My Girl (마이걸)
-Save the last Dance (마지막 춤은 나와 함께)
-My Name is kim Sam Soon (내 이름은 김삼순)
I've watched 3 more dramas, but they weren't so good so i won't recommend it here.
Note:
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Winter Sonata, has played a major part in Hallyu (Korean wave) and made their main actors a huge star in Japan and other countries. This was the 2nd drama i watched, and i have to say that after watching it I decided to start to learn korean.
Very Good Drama and OST.
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Beautiful Days: Very good drama and Very good OST also. It is one of the most popular exported Korean drama series.
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Full House, Goong, My Name is Kim Sam Soon and
My Girl had High Ratings. These Dramas are so Funny and Very Good!
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All In: this drama is very different but Very Good.
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Save the last dance for me: good drama.