Data: eSIM or a physical SIM

If your phone takes an eSIM, buy one before you fly and switch it on as you land. No queue at the airport counter.
If it does not, buy a SIM at a carrier booth in arrivals. Bring your passport; plans usually run from a few days to a month.
Decide first whether you need a Korean number. Data-only plans often come without one.
Even with a Korean number, ticketing sites will usually still refuse you — their identity check wants a line registered to a resident.
Free wifi is on the subway, on buses, and in most cafés and convenience stores. Much of the public network opens without a login.
KakaoTalk is the messenger Korea actually uses. Install it before you go if you might need to reach a guesthouse or a restaurant.