Nodame Cantabile (music, comedy, romance) - It's about two students in a music school.
GTO (1998) (comedy, school) - It's about a former gang leader who becomes a teacher.
Long Vacation (romance) - Minami's fiance runs away on her wedding day, so she moves in with her fiance's roommate.
1 Litre of Tears (terminal illness, romance) - It's about a 15-year-old girl with a terminal illness. This is the highest rated J-drama on MDL.
All of these are widely regarded as classics, so I think you can't go wrong with any of them.