This page is a list of free ways to study Korean, whether you are in or out of Korea.
Learning Hangul
Hangul is fairly easy to learn quickly but it will still take some time to master. There are plenty of resources on the internet for learning hangul, including on the Galbijim wiki. A friend would probably be very happy to help you out with it, too.
Once you've gotten the basics down, it's time to start practicing. And while practicing on vocab sets or flashcards is good, it's even better to practice with 'live' Korean as she's actually wrote. If you live in Korea then you're in luck -- hangul is all around you! Whether you're on the subway, at home, or just sitting and eating some kimbap, there should always be something you can try out your reading skills on. The menu at the kimbap store, advertisements on the subway, store signs, the instructions on your cooking range -- whatever it is, read it. If you're watching a movie channel on TV you'll notice during commercial breaks the name of the current movie appears in the upper-right hand corner -- if it's a foreign movie it will probably be written hangulized English. And when that movie comes on, try to read the subtitles -- that's great speed practice. Also, on the subway in the mornings you can pick up one of the free newspapers they hand out and practice with that -- beats looking at the same advertisements all the time.
If you're not in Korea, don't fret. There are plenty of free resources for you, too. With access to the internet you can head over to Yahoo! Korea, or Naver.com, or Chosun.com, or a hundred other portal sites where you can practice hangul all day long for free.
Just because in the beginning all you can do is sound it out without understanding doesn't mean it isn't good for you. You'll be getting used to the natural flow of Korean syllables and how they go together.
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Learning Basic Grammar and Vocab
There are free resources on the internet where you can learn some basic grammar and vocab. The Galbijim forums are always available for anybody's questions. There is also the free collection of online lessons maintained by Sogang University.
If you have a Korean friend who wants to learn more of your language you can set up a language exchange. Or if you have a foreign friend who also studies Korean, you can practice speaking with each other. Chats between two second language learners can often be more effective than if they each had chatted with a first language speaker.
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Intermediate to Advanced Korean
To practice speaking, you can try going on Skype chat, or another chatting service, with a microphone and headphones and start chatting with other people who speak Korean. Again, the Galbijim forums are there for any question you have or to get some reading and writing practice. Newspapers are always free on the internet. Also, you can read Korea Beat, where Korean news articles are paired with English translations, and you can read the two side by side.
There are also free dictionaries online, though they have a strong tendency to be nearly identical.
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Free classes
If you are living in Korea, especially in the Seoul area, there may be free classes which you can take. One option is the Kongbu Bang, which offers free lessons, focusing on lower-level skills, every Saturday in Yongsan-gu from 4:00 to 5:30 pm. All information including directions is on their website.