Unfortunately you'll need to leave the country to change visa categories and get a work permit - you can see our page on Work Permits for South Korea for more info. I'm not sure whether it's required that you make the application from your home country or whether it can be made from anywhere outside of South Korea, but going to your home country and making the application there would probably be the simplest and best solution.
First answer to Anonymous (07/0417):
What would you like to study? Have you researched which South Korean university might suit your interests? Why don't you have a look at this guide to education in South Korea?
As far as visa requirements are concerned, maybe check out this breakdown of South Korea's visas.
Second answer to Anonymous (12/04/17):
Have you already applied and been accepted for a job in South Korea? If not I suggest you have a look into what working in South Korea will be like, and then apply for the jobs that suit you? What kind of work are you looking to do?
After applying and being accepted for a position, your company should then send you an invitation letter/sponsor your work permit. For more information, see work permits in South Korea.