Prior to December 2010, access to Geoje was limited to ferries or over the one bridge connecting to the mainland at Tongyeong. With the completion of the dual bridges and tunnel that comprise the Busan-Geoje direct expressway link, access by both express bus and car has been made significantly faster and easier.
By bus
Buses leave approximately hourly from the Busan Seobu bus terminal and take 1 hour and 20 minutes to reach Okpo via the new bridge in reality it's actually a bit shorter for 5600 won. Note that where the bus drops you off is often not where you can pick it up again. Services to Okpo for example will simply drop you at a local bus shelter outside the Geoje Museum from where you cannot return. To escape the island, you must make your way to the Gohyeon Intercity Bus Terminal to get to Busan. Alternately, the area around Windy Hill has an obscure intercity link to Masan that runs every 2 hours until 8 pm.
There is also an express bus route from Busan Seobu to Jangseungpo 1 hour and 30 minutes. The bus terminal is a little far from the ferry piers and with no clear directions, so you should take a taxi 2200 won - 5 minute ride.
Intercity buses also still service Geoje from Masan via Tongyeong and also Busan, although this route being now superceded is likely to be phased out taking roughly 2 hours and 1 hour respectively.
By car
Private cars can take the toll expressway from Busan to Geoje. One key advantage to this is that you can take a brief break at a rest stop constructed on a man-made peninsula besides the undersea tunnel, the view from here being stunning.