Agree with what PrettyCarEye said. I'll add some recommendations:
Tokyo Love Story: This is one of
the Jdrama classics. It's popular, and seems to be liked by many, so you might want to check it out.
Lunch Queen: More of a food/family drama (if I can call it that), but there's a romance in it, and the leading lady does have an excellent chemistry with at least two male characters.
Ruler of Your Own World: This drama has so many melodramatic elements rife in early 2000s dramas (fatal disease, social disparity between the leads, dark past, birth secret, love square). But somehow, those elements fit seamlessly into the story that they just felt natural, not cliched. I love the leads because they are so nice and honest to each other, and the acting is superb, too.