When sending mail, the post office in Korea has envelopes that you can buy (this includes bubble wrap envelopes, as I haven't been able to find them at any supermarket). If you require a box, then the post office has a packaging service, they pack your things for you (yes you will pay for this). Or, you can get a box and use their tape to seal it, maybe even get a little bit of bubble wrap from the packaging service. Once you are ready to send your mail, you must get a ticket from the machine (the one with EMS on) and wait for your number to be called. Just like back home, if you post a weird shaped object it will cost a lot more than sending an envelope.
The Korean post website: http://www.koreapost.go.kr/eng/main/index.jsp
Although, finding post cards is difficult in Korea; apparently there is no business in them, and so hardly anywhere sells them. Cards are easy to find, it's just finding the right one for the right occasion.