Hey there, I'm currently taking classes at the Suwon Migrant Workers' Welfare Center. It's very close to Suwon Station. They provide excellent learning opportunities for migrant workers of all backgrounds at an astonishingly low price. 10,000원 for the class fee and 10,000원 for the book. The classes are taught all in Korean since the classes are mixed with a lot of nationalities. However, you can make a lot of progress in a short amount of time. Classes just restarted on March 6th and are currently full. However, in about July, new classes will commence. You can contact them at 031-223-0075 or check out the website. http://www.suwonmcs.com/
However, if you call, be sure to have your co-teacher there because they really don't speak much English at all, more within the Asian languages. They also provide other services so definitely check them out~!
Otherwise, I hear there are also classes at the Youngtong YMCA, but I'm not to sure about them. From what people have said that class is a much more relaxed atmosphere and not as structured. Check that out here... http://www.swymca.or.kr/
This thread from Dave's ESL gives a little information about classes in the past. Or this one but it's really old... http://eng.suwon.ne.kr/notice_view.asp?seqNo=195150
Here's one for last year that I found via facebook. The instructor's number is on there so you could call her and see if classes are still running. http://www.facebook.com/topic.php?uid=25747883752&topic=11768
Here's one for the Suwon YWCA near Jangangu closer to the station. It's all in Korean so get some help from a co-teacher. http://www.swywca.or.kr/sy_sub3.html