Your first day: one T-money card covers it

Buy a T-money card at any convenience store in the arrivals hall and top it up with cash — the card is 2,500–4,000 won, top-ups go in 1,000-won steps.
It works on the subway, buses, taxis and at convenience store tills. You will almost never need to buy a ticket.
Tap again when you get off a bus. Skip it and your next transfer is charged as a new fare.
Transfers are close to free within 30 minutes of tapping off — 60 minutes after 9pm — including subway to bus.
Last subways run around midnight, varying by line. After that, night buses numbered with an N run until roughly 3–4am.
From the airport into central Seoul, the AREX express train is the most predictable. Limousine buses are easier with luggage but depend on traffic.