The most important thing is car access for Jeju. Most of the tourist sites are in Seogwipo on the southern side, almost an hour by car from the airport. There are some places closer to Jeju City and on the eastern side. You must have an international license to rent a car in Korea as a foreigner.
All the resorts are in the Jungmun Resort Complex in Seogwipo. Everything from five star Shilla or Lotte Hotels to more affordable stuff, even glamping and caravan camping by the beach. Caravan camping is often very affordable, especially during the week.
Airfares to Jeju are generally fixed. If you depart at popular times for weekend trips (Friday afternoon, Saturday morning...come back Sunday night, Monday morning)...these will always be around 100 USD each way. Outside of these times, absolutely deals and sales can be found, often just 30-40 each way on the LCCs.