There are many movie theatres in Korea so chances are if you are here long enough you will want to go an see a movie at some time. You are in luck because most of the big American (and some other country) films will show in Korean cinemas in English. They will usually be in English with Korean subtitles, I believe its more common to dub the cartoon ones so watch out for that.
If you want to find out listing times and book your tickets you can do so online at this website http://cineinkorea.com/ - it's an English language service run by BookinBooks Bookstore for Foreigners.
Basically if you want to buy a ticket you send them the money and they purchase it for you (you will need a Korean Bank Account to be able to transfer the money). Once they purchase the ticket they send you a text message on your mobile phone that you can show at the cinema to pick up your tickets.
The biggest cinemas in Korea are CGV, Lotte Cinema and Megabox - you can find a few other ones around also but if you are in a city or town of reasonable size you will have one of these three chains there. The website above covers most cinemas and also has a request part where if you can't find your cinema then you can request that they set it up.
If you are looking for Suwon you will find it under Gyeon-gi Province. There are quite a few around Suwon so you should be able to find one near you. They also have links to maps so you can see the locations of the cinemas listed too.