Are there any teachers, tutors, and/or language exchange partners who don't charge anything (I don't have the means to pay right now), that can help me learn Korean?
http://www.italki.com/partners#learn=english&speak=korean&native=1&page=1
All of the teachers and tutors on italki are paid.
You can find a language partner to do a free exchange with at that link.
Also try interpals.net and lang-8.com