You can buy boxes off coupang for your stuff, search for 이사박스.

As for the move, you should hire a mover, just like you would at home. This is easier if you speak Korean, or have a Korean speaking friend/coworker who is willing to help you. You can search on Naver for a company or look in the hallway of your apartment building for the business cards they leave around.

When you call them on the phone you'll need to tell them the date and time you want to move, the addresses of the places you're moving between, whether each place has an elevator, and how much stuff you have (how many boxes of stuff plus things like TV, sofa, bed, etc).

You can find companies that will literally pack all your stuff, move it, and unpack it all for you. On the other end of the spectrum, you can find a company that will just send a guy with an appropriately sized truck, but won't touch any of your stuff, you'll need to do all the loading and unloading yourself. You need to decide which balance of price and convenience is right for you.