Foreigners or visitors to Korea, who have been in Korea for less than 6 months are eligible for the KR pass (bigger discounts than the Happy Rail pass). Foreigners in Korea more than 6 months can use the Happy Rail pass.
Interested in getting this pass? Go here to book the KR pass https://www.letskorail.com/ebizbf/EbizbfForeign_pr16100.do or here to book the Happy Rail pass https://www.letskorail.com/ebizbf/EbizbfHappy_pr16100.do
Then,you can exchange either pass at a train station (you must bring your passport) for an actual ticket, and you can make your specific rail reservation at the train station.
I used the KR pass last year to travel to Busan for a weekend and I was so impressed with how comfortable and fast the KTX was to Busan. Happy Traveling!