You may want to consider a Working Holiday visa for your time in Korea!
Citizens of Australia, Canada, Denmark, France, Germany, Czech Republic, Japan, Hong Kong, New Zealand, Ireland, Sweden, Taiwan and the United States are all eligible for this program.
The Working Holiday visa is probably the most flexible of the visa options for your time in Korea, allowing you to both work and study. As the rules are slightly different for each country, the best way to find out more information is to contact your local consulate.
