My first japanese drama was Nodame Cantabile ^_^
my first one was bitter blood ^^ 
Nodame fan here too as first EVER drama. got me hooked
It was ,,Pretty Guardian Sailor Moon", which is basically the live-action adaptation of the same name manga and anime :)
My first japanese drama is "Zatoichi the blind swordsman".
I don't remember which season exactly. I watched it very young, we still had black&white CRT (the color one was too expensive for our family) and I think I watched it in late 80's.
Gokusen if I remember well. I had seen pics/gifs of Gokusen 2, which made me interested in jdramas, but then I realised it Gokusen 2 was a sequel so I watched Gokusen first, which I liked very much as well :P
My first was Nodame Cantabile, although the first I truly loved (the second to first I watched) was Hana Kimi.
Hana yori dango because of that I started watching dramas
Great Teacher Onizuka, it was on tv every Sunday morning
ITAZURA NA KISS, ALSO KNOWN AS MISCHIEVOUS KISS LOVE IN TOKYO STARRING MIKI AND YUKI = BEST COUPLE EVER #notashamed #100thtimewatchingit #relationshipgoals #theendingkilledme #idon'tlikethenewcastforthemovie #bringmikiandyukibacktogether 
Liar Game, followed by Itazura na Kiss. Two awesome dramas!