If you are not using taxis or call van services to the airport, either you will take an airport bus and subway/express train to the airport. Easiest route is taking 6102 bus at the formerly Holiday Inn Seougbuk (Now Dongduck University dormitory) or Gireum Station or Jongam Police Station Bus Stops and go all the way to the airport. Or you can take the subway from the Gireum Station and transfer to the Airport Express Line at Seoul Station. Then you will be heading to the airport.
Here is the information on Incheon International Airport Bus System and you will find the info on 6102
http://seoulcity.org/bus-subway-train/incheon-airport-bus-guide-seoul-airport/
and the information on the express train between Seoul Station and Incheon International Airport
https://www.arex.or.kr/main.do
If you have big luggage and you need taxi or van service, we can still arrange it for you if you have never been to Korea before.