I have been Jeju 2 years ago. It is a very beautiful place especially for Honeymoon. Also the expense for Jeju trip is not too much. Jeju Island has a lot of different areas that you might saw from Korean dramas such as Lottle Hotel back garden, most of those areas are free to entry. There is a very special thing I should suggest you to take a look if you have time, meeting “Haenyo” Korean Diving Women - very famous symbol of Jeju.