There are many things to see festivals and events on every Sunday in Spring season at Seoul which is capital of South Korea.
one thing, I'm recommending event is marching of Korean King. This is basically reappearance of Korean king going out from palace. It was big event long time ago. Don't forget to bring your camera. Also, there are many small festivals and events in Seoul, especially, on May. You can get information of events, very easily.
If you like playing computer game, you will need to go Coex and to see game competition, game competition held almost everyday with online broadcasting. Coex is here(http://www.coex.co.kr/index.asp). You will be surprised by many of young girls supporting player in game competition area. :-) almost 10 years ago
The Mud Festival in July is probably the most obvious answer.
Jinhae has a beautiful cherry blossom festival in May. There's also the Jindo 'Moses Walk Festival', the lantern festival in Jinju is beautiful, and there's a few different music festivals around worth a look too.
There are a few more mentioned here:
http://www.aussieontheroad.com/7-things-about-korea-festivals/ over 7 years ago
Busan has two I'd recommend. In the fall, they have the International Film Festival. One of the biggest in Asia, and you can choose from tons of different movies, all pretty cheap.
The Fireworks Festival is pretty fun, too, if you're in the area. Over Gwangali Beach, a huge, 60-minute display of lasers and pyrotechnics. about 6 years ago