Leaving Seoul: KTX or the express bus

The KTX gets you from Seoul to Busan in about two and a half hours. Gyeongbu-line trains leave from Seoul Station, Honam-line trains usually from Yongsan — check which station your ticket says.
Book through the Korail app or website; there is a separate English booking page for visitors, and you do not need a Korean phone number.
Weekend and holiday seats go days in advance. If none are left you can buy a standing ticket, which means standing in the aisle.
Express buses reach the towns the KTX skips. The udeung and premium seats are wide, and luggage goes in the hold underneath.
Bus terminals are not at the train stations. In Seoul the main ones are Express Bus Terminal station and Dong-Seoul Terminal.
The Korail Pass, sold only to visitors, gives unlimited trains for a set number of days. It only pays off if you are making several trips out of the city.