My first trip to Jeju came a few years ago and is a lesson in both not planning trips in groups and then when that fails just saying fuck it and going anyway. Ever since my very first travel experience, which I also took solo due to a friend dropping out last minute, I've enjoyed the freedoms that come with travelling alone. To quote George Bluth, "There is joy in solitude..."
Of course there are those moments of fantasticalness where you want to turn to someone and go "this is awesome" and Jeju has a lot of great places that befits it's a status as a popular tourist destination for foreigners and Koreans alike, most of which have been written about by better writers elsewhere.
However I implore you to stay well away from Jeju Love Land. It is only what becomes of a country that is so sexually repressed and unless you find joy in statues of erect penis and/or elderly Koreans giggling like prepubescent teens at the sight of them then give it a miss.
For me, Jeju's highlight comes in the form of the Jeju Olle trails, founded and literally handmade by former-journalist Suh Myung-Sook and her brother, the trails take in a lot of Jeju's breathtaking and oft hidden beauty. During my two trips I have only taken a small percentage of its routes but I dream one day of taking a few weeks and doing the entire course, wonderfully, all the routes are connected.
The brilliant Ken Lee over at Seoul State Of Mind has done a couple of posts about the Jeju Olle Trails that one should definitely look at.
My other favourite part of Jeju is the Manjanggul Lave Cave... Jeju is a (long-dormant) volcanic island and to my mind, walking through a tunnel that once had molten rock flowing through it is supercoolsuper.
PRO-TIP: The frequentness and cheapness of flights to Jeju from mainland South Korea mean that it can easily be done on the weekends...I would recommend avoiding Jeju during the Korea's holiday periods, Chuseok (thanksgiving), Seolnal (Lunar New Year) and such...pick one or two weekends, even if you are only in Korea for a year, and go then. But definitely go.
How to get there:
Get on a plane...
Get to Jeju City Bus Terminal
From there buses to all places in Jeju are frequent and numerous.
Soundtrack:
Jeju's finest Japanese/Korean electronica hippies have just released an album of ambient drone bliss befitting their chosen home.