Hmm I think I would start with movies rather than dramas since it's shorter and more likely to draw them into the genre (if there is any interest that is). Then, I would indeed adjust to the taste of the person you want to show the world of asian dramas to. If you don't know, I'd go for a comedy or a highly acclaimed/famous movie. eg. My Sassy Girl, Oldboy, Vengence trilogy, Departures, Rurouni Kenshin,...
Good luck!