You can take a train from Busan to Gumi via Dongdaegu Station in around 1 h 40 min.
How to get from Busan to Gumi
From Busan Station train to Dongdaegu Station then train to Gumi Station. 1 h 40 min ₩26,662. From Busan Station train to Gumi Station. 1 h 57 min ₩24,639. From Busan taxi to Gumi. 1 h 45 min ₩169,904. From Busan drive to Gumi. 1 h 45 min ₩25,267.