Gyeongju is a very quaint town, with a downtown area with several tourist sites (parks, a museum). There is a very nice temple and grotto at the top of a mountain, but it's kind of far (30-40 minutes by bus from downtown, but it's an easy bus. Just hop on, hop off.).

Jeju is a beautiful island. Lots of natural beauty there. I recommend taking the ferry over to Udo, renting bikes and exploring for the afternoon. There are many nature tourist sites, as well as a few beachers. By October, it will be too cool to swim.

Gyeongju is definitely navigable by foot/bus/bike rental. Jeju, like I mentioned previously, will require a car or (expensive) taxi rental.