You can get to COEX easily using the subway.
Select your origin and it will tell you how to get to COEX.

From Gangnam Express Bus Terminal      From Nambu Terminal
From East Seoul Bus Terminal    From Gangbuk (center of Seoul) area

From Nambu Terminal
Take the Subway Line #3 at the Hwamul Terminal (Cargo Terminal) station and get off at
Gyodae (Seoul National University of Education) station. Transfer to the Subway Line #2 in the direction towards Jamshil and get off at Samsung station. Use the Exit No. 5 or 6 connected to the COEX Mall.