About learning through dramas...
I seriously learnt a lot of words through dramas.
But when I watched my first drama (상두야,학교가자!) completely (not only a few episodes like I did when I had to rent VHS tapes from the food shop...) I was already in a lower intermediate/upper beginner level.
Yet, even if you're not I think listening to a language more often makes you get used to the sound/melody/stresses of the language. If you can "feel" where one word ends and the next starts (you couldn't in a language you don't know) it already helps. If you get a feeling for the way ppl express their feelings with the melody etc.
Just like Matthew said with time I recognize more and more vocab, grammar patterns etc. and if I don't I just enjoy the show.
Often, though, they talk about a specific topic and repeat the vocab for it, then I try to write it down in Hangeul and in English, then later I go check in a dictionary. If I don't write it down I might forget the word again.
The words I mostly learnt through dramas are words that describe relationships, feelings, titles (like section chief etc.).
to marry is one of the words I learnt through dramas
etc.
"get out of the way" another
By now I already know some words around dating and marriage in Japanese hahaha