It's really easy here, you just need to buy the special bags from all supermarkets (Emart, Lotte Mart, Home Plus) and most groceries stores...they cost about W6 000 for a pack of 20 and come in loads of different sizes.
*NB* Make sure to get the bags for YOUR area, as there are separate bags for each district. Here are some pics of the green recycling bags for the district I stay in, Yuseoung:
South Korea is extremely mindful about recycling, to the point where there have been some rumors of harsh fines! So stay out of trouble and recycle your waste :)