Drama is a very vague word for me. Do you mean Live Action Japanese Movies or Live Action Japanese TV-Series? If you want to try movies I would normally suggest something by Akira Kurosawa but he can be pretty "heavy" at times, with the exception of his movies "Yojimbo" and its sequel (of sorts) "Sanjuro" both of which are old Samurai classics movies. Both these movies have always been available (subtitled) on video and DVD. Hope this helps.
In the TV-series department however I can't help you.