Daegu has two train stations.   Daegu Station is located just off the side from downtown.  Unfortunately Daegu Station is NOT served by 1st class trains.  Dongdaegu Terminal is served by all trains which enter and exit Daegu. 
 

Daegu Station         : 425-7788 Dongdaegu Station : 952-7788

ΆΖ KOREAN NATIONAL RAILROAD - http://www.korail.go.kr/eng/index.html

 

	Daegu and Dongdaegu Station are on the subway lines and the subway stops are named after the train stations.    Since Daegu is on the major North-South line, any train going to/from Busan and Seoul goes through Daegu.   In addition Dongdaegu Station offers trains on the less frequently used East-West line.
Remember that weekend train journeys usually require either reservations or standing up on the train.  If you want to make reservations but don't have time to visit the train station, you can buy a ticket at the nearest Gu-office (you know, like Namgu, Seogu...).  You can also purchase airplane tickets at the Gu-offices.