First, Incheon airport-Seoul/Gyungki.
Train. Express train from terminal (T1) or T2 to the Seoul station. official site of express train is here. The best way to the every spots in Seoul.
Bus. It you have to go to the suburb like Gyungki. Bus may better but the reservation is a little bit tricky. However, please don’t be afraid because it serves English language. Official site is here. You can buy the tickets in the ticket office but it usually has a long line. You can not buy tickets inside the bus so the best way is make a reservation in advance.
Taxi. If you are together, may be it’s cost effective but they usually carry luggage up to 2 or 3. To the central Seoul, it costs about 50.000W without traffic. Buses are even more comforter when the traffic is heavy. Sadly, Uber is unavailable in Korea.
2. Second, Airport-local provinces
It depends where it is. Cities usually have a KTX train station. Basically, from Seoul station you can go every where if it is a city. (also Wonju)
If your destination is not a city, buses are the best way to go.
3. Buses and subway
Generally, Korean transportation ticket service offered by T-money. You can buy Tour card by T-money every convenient stores named CU, GS 25, seven eleven. I recommend you drop by seven eleven in the airport (1st floor) and buy below transport card (design sometimes can be changed). Can charge with cash only.
Official site of tour card is here.
+ Metro app
Kakao metro : It serves map, arrival time, transit place, time to get the destination. Can not buy tickets or pass through the ticket machine with this app.
+ Map app
You can use google map in Korea but the even the latest version is not following current roads in Korea.