Major internet providers: KT, LG, SK
Internet/TV/phone/mobile service is usually from one of the main three telcos: KT Olleh, LG U+, or SK Telecom. Discounts are given for multiple-year contracts, or for combining several packages (for example, internet+mobile).
A 1-year Internet contract from KT/LG/SK will cost around 35,000 won a month. Keep an eye out for promotional flyers around your building - you can find promotional deals for 22,000 a month.
Unless you are absolutely sure you are staying for 2 or 3 years, do not sign up for a 2- or 3-year contract. You can technically transfer your contract to a new person, but there is no guarantee you will find someone.
Smaller (and slower) internet providers
CJ Hello Vision provides service to Bupyeong-gu and Gyeyang-gu. A 1-year contract costs around 13,000-19,000 won a month, depending on what kind of promotion is currently going on. Wifi routers are only available to 3-year contracts, but you can easily buy and install your own wifi router for around 25,000 won. Transferring service to another location will incur a 22,000 won installation fee.
Wifi Routers
Your internet provider may or may not provide you with a wifi router. Sometimes you get a modem and have to buy a wifi router for it, sometimes you get a modem+wifi router.
Buy a English-language wifi router online or from one of the big marts (Homeplus, EMart, Lotte Mart), for about 25,000 won. A widely-available brand with English firmware is TP-LINK. Korean software is probably pre-installed, but you can easily overwrite this by downloading the English firmware from the manufacturer website's support page for your specific wifi router model.