Skip Beat (However was not in Japanese but based on manga and anime)..Seeing the live action was wonderful, then I found more dramas and love all the romance
Proposal Daisakusen
First K-drama - Full House
First J-drama - Buzzer Beat ^-^
Nobuta wo Produce
First K-Drama is You're Beautiful, a great way to start watching K-Drama, it was amazing.
First T-Drama is Sotus the Series, i'm completely in love with it!
boys over flower
My cousin was the one that open my eyes about the beauty of Asian dramas and Kpop.
She started me with One liter of tears and we followed with Werewolf boy (movie)
But once she left (she lives in another city and she was visiting me) I started on my own with
You're Beautiful.
My Princess in 2011, but the drama which really got me hooked in the dramas sphere was Coffee Prince.
My first drama was Marmalade Boy the Taiwanese version. 
However my first Japanese drama was Hana Kimi. Never looked back after that.
But, gosh I wish they would make a Japanese version of Marmalade Boy.
My first drama was Descendants of The Sun
My first JDrama was 1 Litre of Tears
 Dream High 2011.