My first stop in Korea after the airport was the Kimchi Hongdae Hostel, which would be home for at least four nights, on the first draft of my itinerary. Situated in Hongdae, the hostel is around a 10 minute walk from the Hongik University subway exit 2 on either the AREX commuter train or the green outer circle line, making it an ideal location.
Hongdae is the slang term for Hongik University so don't try to look for Hongdae in the subway map as you won't see it! This area is known for the all night bars and clubs where some of the younger crowd - Koreans or foreigners alike flock to.
The area is also known as a shopping heaven, full of indie stores (except for a few chain cosmetics stores that you would practically see everywhere) that offer cheap clothes, shoes, or accessories that's of a very good quality.
So, why did I choose this hostel? Aside from it's proximity to Hongdae and it's accessibility to the subway and the airport express trains, they offer cheap dorm rooms.
They're also one of the cheapest hostels around. Online, they have great deals for rooms. As of this writing, if it's a weekday, you could get a dorm room for 15,000 Korean Won (roughly 15 USD), and 18,000 Korean Won (18 USD) for weekend rates.