Japanese: Pretty Guardian Sailor Moon (2001) - it is the reason, why I got to know J-Dramas and K-Dramas in general.
Korean: Sweet 18 (2004) - I stumbled across this show by accident - while searching for PGSM.
Taiwanese: Marmalade Boy (2001)
I watched these around 2007/08... I think....
I do not watch a lot of Chinese Dramas, so the first is easy to pin down: "Let's go, watch the Meteor Shower" (2009). (Don't know, when I watched it...)
My first Thai-Drama was "Full House". I tuned in when it came out in 2014.