Inchon happens to contain Korea's international airport. It is a fairly nice airport that is similar to other major world airports, including overpriced stores/restaurants.
Here is the English-language link to the website: http://www.airport.kr/eng/airport/.
Be sure to arrive early as the check-in lines can be quite long (most Koreans show up about 2 hours before an internation flight -- try to show up earlier so you don't have to rush through security and to your gate). It takes a while to get to your terminal (trams and stairs/escalators are involved).
There is internet access on the second floor outside of the arrival area. I think there are a few internet terminal areas in the departure area behind the security gates but they are far more expensive and can be crowded.
There is an AREX train/shuttle (the express is similar to a train while the slower transport is similar to a subway car) that goes from the Inchon Internation Airport and costs 3,100 Won as of 9/3/08. The AREX line stops at Inchon Int. Airport Cargo Terminal, Unseo, Geomann, Gyeyang, and Gimpo Airport in that order and takes about 30 min. (express) to 45 min. (regular). There are plans to extend the line to Seoul Station. From Gimpo Airport, you may access the Seoul subway system (it is difficult to find a map in English; you can grab a map in Korean and use the numbers to navigate your way around). English-language subway maps may be found in Itaewon (the tourist info. room) and at an Indian food restaurant in the same area. (http://www.nsubway.co.kr/korea/seoul/seoulsubwaymap.htm
This website contains an English-language version of the subway map: http://www.workesl.com/bbs/board.php?bo_table=emember&wr_id=22.
I advise you to grab English-language materials when you can because they dissapear quickly and may not be reordered for a while.
A) I strongly advise against using Asian-based carriers for multi-destination stops. I have had to fight for a seat one too many times when my flight was delayed and caused me to miss a connection. One particular carrier always claims to have a full plane until you pull out extra money. I had to get a manager of the airline responsible for my delay to book me a seat as the general ticketing agent didn't have access to this carriers computer system. In other words, not all Asian countries are as service-based as Westerners expect.
B) One Korean carrier's English-language website will not allow one to purchase air fare. I wished I had known this sooner. I ended up giving the credit card number over the phone after finding someone that could speak English.
C) Korea is not an incredibly travel-friendly country if you speak English and go beyond a few select cities (Ilsan is great). You definitely want to learn a few basic phrases if you travel outside of the Seoul and Busan areas that are not near subway lines. Here's a good free site to help you with that: http://rki.kbs.co.kr/learn_korean/lessons/e_index.htm.
D) Occaisionally, there is anti-American sentiment, so pay close attention to the political situation before travelling to the Republic of Korea.