From what I've found, there's no direct air from Busan Gimhae airport (Korean Air, Asiana, Air Asia, etc.)
To fly, one must connect to Yeosu from Seoul. The airport is on the other side of the bay from the Expo. Takes a short taxi ride to get to the Expo.
There is rail and bus service.
There is the regular 'slow' train that takes about 4.5 hours. On this train, one must transfer to Yeosu at Suncheon, a nearby town. Pay for the KTX from Suncheon to Yeosu. It's cheaper, too. There are no English instructions when the train needs to change tracks so just be brave and ask.
For high-speed KTX, there is no direct line. from Busan. However, if you want to travel in style and are feeling adventurous, anyone can take the KTX to Daejon and then transfer down on the newly opened Yeosu line.
I did find that most bus terminals will run service to Yeosu Bus Terminal. As per usual, most bus station websites aren't set up for online reservations. One must simply arrive at the station and hope that their are tickets or go directly to the terminal in advance to buy.
Nopo-dong Central Bus Terminal (No online time table, reserve at station):
Nopo-dong West Bus Terminal:
**Note that there are two different terminals for Nopo-dong at the same location. Can't remember if they are attached, however.
Here's a sample schedule departing from Seobu Bus Terminal (Sasang Stn.) (3.5 hours and count on traffic jams there and back.)
I have no info on ferry service which might be available.
If anyone has additional information or has read anything misleading here then feel free to post.