If you go to Korea, whether for a long stay or short stay, don't forget the Korea Rail Pass, especially if you're between 13 and 25 or travel with 1 or 4 other people! With one pass, you can travel around the country by KTX for unlimited rides for a certain period (depending on how many days you buy). You can buy a 1 day pass, 3, 5, 7 or 10 days pass.
For a weekend (3 days 2 nights) in Busan, this is the best solution : it costs 67,700/person for 3 days of unlimited train. That's cheaper than the regular round trip from Seoul to Busan!
You can apply for the pass online, print the receipt and then go to one major Seoul station to get your pass. Make sure the dates are right when your pass is issued!
Then, at the information counter (or tourist office, depending on stations), get your train tickets (seat appointment) by showing your pass.
When you ride the train, you should have your KR Pass, your train ticket, and your passport! When I travelled without my passport, it was ok though.
If the train is full and you can't get a seat appointment, you can still ride the train! That's the best part of the KR Pass. You can get a standing seat and only show your KR Pass. :)