Yeosu Airport is useful for both Yeosu and Suncheon, but the onward trip is different enough that you should decide your destination before leaving the terminal. Korea Airports Corporation currently places the Yeosu/Suncheon city-bus stop about 30 meters to the right after exiting Arrival Gate 2; the taxi stand is across the road from the same gate.
For a quick sense of distance, the airport's current taxi guide estimates about 20 minutes to Yeosu City Hall, about 20 minutes to the Yeosu Station/Odongdo area, and about 20 minutes to Suncheon Station. The listed sample fares are KRW 12,000, KRW 20,000 and KRW 25,000 respectively, but the airport explicitly notes that traffic and arranged rates can change the actual fare. Outer destinations are a different proposition: the same official page estimates roughly 60 minutes to Hyangiram, so do not treat every Yeosu attraction as a short city transfer.
If you want the bus, check the current service on the travel day rather than saving an old timetable. The airport's official contact page lists separate operators for Suncheon-bound and Yeosu-bound city buses, which is a useful reminder that they are not one interchangeable route. For a late arrival or a tight hotel check-in, keep the taxi stand as the fallback.
Before landing, save the Korean name of your exact destination, confirm whether it is in Yeosu or Suncheon, and check the final bus connection. If your destination is a ferry terminal, island connection or outer attraction, build that as a separate second leg rather than assuming the airport transfer reaches it directly.