Here is a link to site where you can buy modern-style cloth diapers in Korea. You can choose from velcro or snap closures. The website is all in Korean, so if you may need help, but they look good. It looks to me like the outer cover is similar to Fuzzi Bunz and Bum Genius style covers, but you don't put the insert in the pocket of the cover, you lay it in so it is next to baby's skin. I thought I would like velcro better, but I prefer snaps. If you are just starting with cloth, get some of both in the first size and see what you prefer. I think the snaps look better and hold up better. My velcro is still going strong, but I still prefer the snaps.
Here is a link to buy Kissaluvs in Korea. Kissaluvs require a cover as well. I prefer the other style.
Sarah and I plan to go look at these in the store. I'll bring along some of my Fuzzi Bunz Mediums and Bum Genius One-Size for size comparisons. I can't tell the cost from the website whether you are getting the same number of covers and inserts in the package deals or whether it is two covers and a lot of inserts. That is what the picture looks like.
When we go, I will update this post with the location of the store, directions to the store, size and price comparisons. There is a Fuzzi Bunz sale going on right now so I might buy some size smalls for the new baby. Ian's medium Fuzzi Bunz won't fit much longer. Hopefully, there will be some Korean cloth diapers in bigger sizes that he can wear or we'll just use the Bum Genius diapers. Those look like they will fit longer.
I've been told that you can find the old-fashioned prefolds at places like Emart, but I haven't confirmed that.