My teacher told me a story about one of her students, who was at the time 20, moved to Mexico with no knowledge of Spanish, but after a year of living there she came back being able to speak Spanish fluently.

Now I'm wondering if my mom speaks to me in Korean from now on so I can learn the language, how would I learn the grammar, vocabulary, and such? Would I be able to infer what she is trying to say to me over time if she keeps repeating it?

You probably wouldn't learn very quickly or effectively. That approach would have yielded much greater results if you were about 1-4 years old ;).

The other person completely immersed herself in Mexican culture. When the brain has no choice but to adapt, it can often adapt very quickly.

As everyone says, the best way to become fluent in a language is to drop yourself into a foreign land.

Age also plays a factor. I read somewhere that the average eight year old can become more or less fluent in a language within 2-3 months, without formal training (of course, it depends on the language -- obviously a Romance, Germanic, or just plain Indo-European language would be easier to attain than, say, Thai). As you age, it takes a lot longer to learn not just languages, but new information in general.

Oh ok, now I understand a bit more.

I have another question, should I learn the alphabet first and then move onto vocabulary?

what was your first word, "abc" or "mommy"?