According to the city government, you can get to anywhere in South Korea from Daejeon within half a day.
Seoul and Busan are less than an hour and two hours away, respectively, on the KTX. Gyeongju is also now rather accessible courtesy of the new Shin-Gyeongju KTX stop between Daegu and Busan. Seo-Daejeon Korail station will similarly link you up to Gwangju.
There are several pick up spots around town for the airport bus, which charges a princely sum of about 14,000 won for the ride 3,500 won for the regular inter-city bus. Also see this articles Get in section for information on travel to Jeju. Another option to get to Jeju would be to take KTX to Busan and catch a ferry but this would be both less economical and efficient.
Closer places of interest include:
Mt. Daedun, one of Korea's more famed mountains you may have seen photos of its suspension bridge is closeby at a 45 minute bus trip away. Buses depart from the Dong-gu Intercity bus terminal at 10:30AM. Miss it and you're out of luck.
Gongju, adjacent city and former capital of Baekje, known for its Beakje tombs. Also the gateway to hiking a nearby national park.
Cheongju, Daejeon's airport, and historic town in its own right. Birthplace of jikji, the world's oldest moving metal type print book.
During festival season, Daejeon is the natural transfer point for travellers from the south end of the country heading to Boryeong for its expat-oriented mud festival.
