The other big cities are Busan, Daegu, Daejeon and Gwangju. Also, being in these cities for two months, you'd probably need to learn some Korean to help.

Do you actually need two months here? Where did you want to go? Most small tourist cities and other towns can be seen in just a day and a half or two - Gyeongju, Damyang, Danyang, Andong, Sokcho.