Hi, nice to see you in busan!(i'm korean, so my english might not be good..)
you can buy nice pair of shoes from internet. fiveten, la sportiva, madrock and etc.. is available. it's quick, one day after you'll get it.
If you know your size well, then It will be very easy to order..
If you want to wear it before buy it, Since Not so many climbing gear shop in busan. so your choice is very limited..
Mack (intoutdoor) knows where the gear shop is.
However I'm adding a map for your own.
Since foreigner's feet are bigger than korean, It might be hard to find the shoes which fit you well.
for the Gym, better choose nearest one. Cuz most of gym, better don't expect they can speak english well. but most of them are friendly to foreigner. So don't worry. they wont harm you.
if you want to do some bouldering on weekend then tell me.
I'll show you what we have been doing in busan.