Booking accommodation online
As mentioned elsewhere, I have, with a couple of exceptions, opted against mentioning specific hotels, hostels and guest-houses on this blog because information gets outdated, places close; deteriorate; improve and prices change. Instead, I would rather direct your attention to booking websites where you can read recent reviews from people who have stayed, and also get access to information about hundreds of different sleeping options that might better serve your needs.
The best impartial source of information is probably Trip Advisor; a collection of reviews from guests, a ratings system that puts all hotels in order of quality based on those reviews, and also provides multiple links to websites where you can reserve your room. It also allows you to narrow down your search depending on your requirements. Refined searches on this website even allow you to single out accommodations with a shuttle bus service, room service, wheelchair access etc.
Trip Advisor is more useful for top range hotels, than it is for budget options. If like me, your concern only goes as far as having a roof, walls and a bed in the place you sleep, then there are hundreds of hostels now in Korea, and more are appearing all the time. Try the following websites for booking: hostelworld.com, hostelbookers.com, hostels.com and of course YHA/HI hostels.
I would say that Hostelworld has the best coverage and, as it is the most used, has better reviews. YHA/HI is an international brand, so can be more reliable. Prices vary. If you buy a membership to HI then you will also be able to stay for a cheaper price. Generally every major city in the world has an HI affiliated hostel and breakfast is usually provided in the price.