Here are a few spots in and around Jeollanam-do where you can admire the fall foliage.
Baekyangsa temple (백양사). Located on Baekyangsan mountain in Jangseong county, one of the most popular local destinations for looking at 단풍. If you were in the same orientation group as me in 2006, you'll remember we visited this spot. Last year they had a festival on November 1st and 2nd, though the page says nothing about this year. Here are some pictures from Naver; this is my favorite one. Buses run to Baekyangsa eleven times a day from the Gwangju Bus Terminal: 6:35, 7:30, 8:40, 10:05, 11:20, 12:45, 14:15, 15:20, 16:50, 18:15, and 19:50.
Daewonsa temple (대원사). Located in Boseong county, this is one place I'd really like to visit. It's especially beautiful in the springtime, though, and has a Tibetan Museum. Some pictures available here from Naver. This is the official site, which says buses go to the temple from Beolgyo-eup three times a day: 8:30, 1:20, and 4:30.
Gangcheonsa temple (강천사). Located in Sunchang county, Jeollabuk-do. Buses make the ninety-minute trip to the temple from Gwangju terminal ten times a day: 8:10, 8:50, 9:50, 10:30, 11:25, 13:30, 14:10, 14:50, 15:30, 16:10. Here are some pictures from Naver.
Mudeungsan (무등산). In Mudeungsan Provincial Park, the largest mountain in Gwangju and one of the largest in Jeollanam-do. There are a number of local buses that go there: one way to do it is take a bus to Jungshimsa, walk up the mountain for thirty minutes until you come to a paved road lined with maple trees, and follow this road to another temple, Wonhyosa. I'd be interested to hear of some other routes from people familiar with the mountain. Some pictures from Naver, and here is a nice gallery from a blog.
Naejangsan (내장산). Located in Naejangsan National Park in Jeollabuk-do, this is one of the most popular, and thus most crowded, spots in the country to see colorful maple leaves. The pictures from Naver look absolutely gorgeous, though, especially this one. Buses run to Naejangsa temple five times a day: 8:15 10:25 12:20 13:40 15:05. The park's official site says to take a bus to Jeong-eup, and then take local buses 171 or 171-1 to Naejang terminal.
Piagol Valley (지리산피아골계곡). Located in Jirisan National Park in Gurye county. Buses run regularly from Gwangju and Suncheon to Gurye, from whence you can take a bus to Yongoksa temple. I visited the area in October, 2007, and it was a much more leisurely hike than I was expecting. I walked a few kilometers to the first rest area, then turned around.
There was a festival scheduled this year, but it was one of many Jeollanam-do ones cancelled because of swine flu.
Samseonggung (삼성국) and Cheonghakdong (청학동). In Hadong county, Gyeongsangnam-do. Hands down the most beautiful and striking place I've ever been in Korea. This post I did last year has some more information, and below are some pictures:
To get there, take a bus or train to Hadong-eup. Then, take one of the five buses a day from the terminal to Cheonghak-dong: 8:30, 11:00, 13:00, 15:30, and 19:00. When you get off the bus, walk up the hill to mosey around Cheonghak-dong, then go all the way back down the hill and over to Samseong-gung. I'd recommend getting one of the earlier buses to Cheonghak-dong, because you'll want to give yourself plenty of time to see both sites.
* Update: Here's some information about peak times for viewing around the country, from this An Acorn in the Dog's Food post.