Learn the Korean alphabet. The alphabet was invented in 1443 during the reign of the Great King Sejong. There are 10 basic vowels and 14 basic consonants. Letters with similar sounds also have similar shapes.
Come up with a list of Korean songs that you like. Popular Korean songs include "Nobody" by Wonder Girls, "Hug" by TVXQ, "Gee" by Girls' Generation, "Sorry Sorry" by Super Junior, "Lies" and "Day by Day" by Big Bang and "Because I'm Stupid" by SS501.
Find the lyrics of your favorite Korean songs as well as a line-by-line English translation each. Learn the words that make up the songs. Listen to the songs as many times as you possibly can, trying to discern separate words, phrases and sentences.
Learn the Korean songs by heart and try to sing them. Learning whole sentences will help you build a rich vocabulary and allow you to memorize the context in which the words are used. Because the phrases and sentences are rhymed, they are easier to memorize than non-rhymed sentences.