Beautiful days.  After that i started scouring for Takaya Kimura productions...I have watched most, not all are great works but overall i think he is a good actor.
Mine was a veeery sad drama 'One Litre Of Tears' I didn't expect it to be that sad I just randomly started watching it bc it was feautured on a site T^T
Hana yori dango :)
"Ghost friends" :)
Hero (with Takuya Kimura)
Not sure how old you need to be to remember these shows, but I'm almost certain the first J-Drama I watched was Abarenbo Shogun. If not then it was Oshin. It was definitely one of the 2 though...lol
It was Akumu-chan, I watched it for Gackt, but never finished it...
It was Mother. I fell in "squee" with Ashida Mana. I am not sure if another child actress would have left such a lasting impression on me. 
Like for many people here it was Hana Yori Dango for me and I've started to pay more attention to Oguri Shun. ^_^
My first drama was Hana Kimi as I received it on DVD, which then got me into other JDramas :)
1 Litre of Tears.