Daegu and Gyeongju are quite close to each other; I'd say about 45 mins away on a good day. Because of the proximity, there's always a bus leaving maybe in 10-min intervals.
By train, I mean regular service. I'd say the trip is maybe 35-45m? However, I haven't taken a train in such a long time b/c you'll have to catch it at the exact minute. It's been my luck that buses were always available.
In an orderly nut shell,
- the easiest, but most expensive method is KTX Seoul (surrounding area)- Gyeongju.
- Next, reg. train or bus from Gyeonggi-do to Daegu. After, bus Daegu - Gyeongju
I live in Pohang, the easternmost city, so I'm usually taking trips to Daegu and Gyeongju.