The very very very first one, was
OSHIN. I watched this together with my mother when I was still a little kid. Although I only remember the gist of the drama and not the complete story (mind you, I watched this more than 20 years ago), but it will always hold a dear place in my heart.
And then around 10 years ago, I watched
Friends-2002. But I can't really categorize this fully under Japanese drama, because it's a collaboration project between Japan and Korea. Won Bin and Kyoko Futada were the main leads.
So if I think about it, the very first (fully) Japanese drama that I watched when I was an adult was
Boku dake no madonna with Takizawa Hideaki and Hasegawa Kyoko. And the rest is history.