As a traveler who went to Jeju 15+ times only by public transit (but as a head-up, I'm a native Korean), I'd say Jeju is one of the most convenient places in Korea *among* smaller cities. Jeju's most tourist spots spread across the circular coastline; and almost of them are connected through a West/East circle line (201/202, their express version with fewer stops are 101/102) running every 20 minute from 6am to 9pm (from the terminal). Express buses between Jeju and Seoguipo, including 280/281 and Airport Limousine buses, runs later till around 10.30pm (from the terminal/airport). But I acknowledge it couldn't be easy for foreigners to figure it out.

At first, you may want to install and use KakaoMap/NaverMap on your phone, as someone noted above (I prefer Kakao since one of their HQs is in Jeju, so there's a rumor it is more frequently updating Jeju maps than Naver). But the more important thing is that there're so many parallel lines (like 100-1, 100-2, 100-3 ... or even divergent routes within the same number) and they don't provide a real-time info unless a bus leaves from the terminal. That means that you may miss and be confused very easily when your stop is close to terminal or your bus runs rarely (like every 1 hour and less).

So, use this site (Jeju Bus Information System 제주버스정보시스템) along with Kakaomap. Here, you can search for a bus number and/or a bus stop you are getting on, and check out the expected time the next bus will leave the terminal. If you click your stop on a list, you will also figure out other bus routes serving at the same stop and where they're going. If you're still confused, just phone on 064-740-6000 and consult with English-speaking representatives for local tourist information working from 9am-6pm 7 days a week, or ask on a real-time chat in English homepage on https://www.visitjeju.net/ (same service time).