Just looking at the subway system map can make it seem daunting but in fact it is relatively easy to navigate. Lucky for us most of the signs are in both Korean and English and the pre-recorded Subway Station announcements that tell you the station, any transfers and which side to exit are all followed after the Korean by the English translation.
The subway connects almost all destinations of Central and Greater Seoul and due to heavy traffic on the roads it can often times make travel faster.
The cost is minimal, around 1000-2000 won and you can buy your ticket at the stations using the ticket machines (which have English). You can also purchase a rechargeable transport card such as T-Money - these have the added bonus of being able to be used on the buses and some taxis and they also offer a small discount on ticket prices.
The metro stations are one of the best landmarks for navigation in Seoul. You will find that if meeting someone it is easiest to arrange a meeting at one of the stations as you can designate the number exit to ensure that you are both in the same place.
There are a few websites that you can use to plot your journey, they usually will tell you the fastest way or the way with the least amount of transfers. There are also some smart phone apps that you can use that do the same thing.
Jihachul - Apple and Samsung
Seoul City Metro (Lite version for free, Pro version costs) - Apple
Korea, Seoul Metro Navi - Samsung
Suwon is on the Dark Blue - Number 1 line. There are four stations that service the Suwon area - they are Sungkyunkwan University Station, Hwaseo Station, Suwon Station and Seryu Station.
Suwon Station is the main subway station for Suwon, the station itself shares the building with AK plaza which is a large shopping mall. There are many shops and restaurants both in and around the station and there are buses and taxis available right outside the station if you need them.