Has anyone ever been to Cheju Island in South Korea?

Many times.
Sometimes you will see it wtitten as Jeju-do. (do is island)
I have mostly gone in the summer, but once in colder weather for the jeju orange/mandarin orange festival.
it really is only warm enough to enjoy the beach/swimming in teh height of summer.
try to avoid the first week of august, when all Koreans are there on vacation and the prices are hte highest.
My favorite area is Jungmun resort area. Hilton is the only hotel with views of the ocean from your room and also ha a small private beach. (which is too rocky for swimming but very romantic)lots of caves, mountain hikes ans such on Jeju.. you could easily pass a week here, so stay for longer than a weekend.
Most people rent a car but I use the island bus system.... not as efficient and timely as Seoul, but I would rather do it the local way. this will require a bit of walking when you go from city to city as the island circle bus stops are few and far between, but once you get to Jeju City, they have a bus system similar to seoul.

no