My very first Japanese experience was a movie: "Tada, Kimi wo Aishiteru". So I can safely say the first words I learnt were Tada (only, just), Kimi wo (you, informal) and Aishiteru (love). I couldn't even begin watching before I knew what the title meant, so I learnt - well, let's say I read about it - the use of particles like ga, wa, de, ni etc.
While listening, though, the words that always stuck to my minds were:
Doshite - why/how
Dakara - therefore
Okatta - all right, what a relief!
and my favourite: "Nani ittendaio?" (or so it sounds), which means something like "what are you saying?"
As for Korean, my first was... My Girl, if I'm not mistaken.
The only thing I remember was thinking how weird it sounded (I was still very much into Japanese mode). I learnt the courtesy suffixes: "-kah" and -mida". Not much to have a conversation. ;)