On foot
Nothing in downtown Jeju is more than approximately 2 km from anywhere else so whilst not convenient, walking is quite feasible, and sometimes even faster. This includes the areas of Yongdam, Tapdong, Jungangno and the inter-city bus terminal.
By bus
Jeju City has an inner-city shin-nae bus system that is equally as effcient as it is poorly marked for tourists. Bus shelters have lists of stops in Korean only, but the stop names are mostly fairly non-descript and require you to have local knowledge or a map to make any sense out of them. Bus maps are not available from tourism centres, but English station lists can be obtained, the relevant information from which is reproduced below:
Jeju International Airport - 36, 37, 100, 200, 300, 500
Jeju Ferry Terminal - 92
Inter-city Bus Terminal - 26, 31, 63, 100, 200, 300, 502
Halla Arboretum - 63, 300
National Jeju Museum - 100
Culture and Art Center - 300
Tapdong - 92
Yongdam Junction - 36, 37, 63, 300, 500
Fifth-day Folk Market - 36, 37, 63
Jungangno - 36, 37, 92, 100, 500
Dongmun Market - 100, 300
Shin-jeju Junction - 26, 31, 36, 37, 92, 100, 200, 300, 500, 502
Halla Hospital Medical Center - 26, 31, 37, 92, 200, 300, 500, 502
If this system was not frustrating enough as it is, you may have to transfer to go most places and the announcements tend to be quieter than on the mainland... except the Chinese one which mistakingly sounds like something somewhat rude to an English-speaker. Listen out for this when having to transfer between buses. Buses cost between 950 and 1000 won.