An extra week will allow you to spend some time getting a better sense of the beautiful countryside and historic sites South Korea has to offer.
Day 1: Arrive in Seoul
Arrive at the Incheon Airport and check into your hotel. Have dinner and see a show at Korea House or dinner with a view at Top Cloud restaurant. If you get a chance, soak in a sauna or bath at your hotel or at one of Seoul's many bathhouses.
Day 2: Explore Seoul
Tour Changdeokgung Palace -- guided tours in English are held Tuesday to Sunday at 11:30am, 1:30pm, and 3:30pm. Then take a walk around the city's cultural street, Insadong, where you may want to stop for a cup of Korean tea or lunch of traditional temple cuisine (always vegetarian, often bland, and usually made with local ingredients) at Sanchon. Then, squeeze your way between crowded stalls at Namdaemun Market. Enjoy a quiet dinner or, if you want a livelier night out, take a dinner cruise on the Han River. If you can't sleep, head for late-night (or early-morning) shopping action at Dongdaemun Market.
Day 3: Take a Tour of the DMZ
Allow a whole day to take an organized tour to Panmunjeom, also known as the Demilitarized Zone. If you're not too exhausted upon your return, head to the top of Namsan and the N Seoul Tower to admire the view of the city lights.
Day 4: Suwon Fortress & Korean Folk Village
Take a subway to Suwon and get a ticket to the Korean Folk Village at the tourist information booth. Free shuttles run every hour on the half-hour. Explore the village and linger over a lunch of traditional food like janchi gooksu ("party" noodles, a simple somen noodle soup with broth and a bit of seasoning), nokdu buchingae (flatcakes made from ground mung beans), naengmyeon (buckwheat noodles in a chilled beef broth), and makgeolli (a milky, unrefined, traditional rice wine), served as you sit at outdoor tables. Take a free shuttle back and catch a bus to the Suwon Hwaseong. Stretch your legs and work off lunch with a nice walk around the fortress. Enjoy dinner of the famous Suwon galbi (marinated short ribs) and then head back to Seoul for a good night's rest.
Day 5: Transit to Gyeongju
Allow half a day to travel to Gyeongju (going by either train or bus takes 4 1/2 hr.). Explore some of downtown's historic sites, such as Tumuli Park, where the Shilla kings are buried, and Cheomseongdae Observatory. If you get a chance, try one of the Gyeongju's sweet bread snacks such as hwangnambbang (bread filled with a sweet red-bean filling).
Day 6: Explore Gyeongju
If you've explored the downtown area the previous evening, pick up something for lunch and take the Gyeongju City Tour Bus to explore points farther out (which are difficult to see individually), including Seokguram Grotto and Bulguksa Temple. Take an evening train or bus to Busan.
Day 7: A Brief Visit of Busan
Finish your whirlwind trip with a morning walk on one of Busan's sandy beaches and an exploration of Jagalchi Market and a lunch of hwae (raw fish). Catch a high-speed KTX train back to Seoul for your flight back home.
Day 8: Extra Time in Busan
A week's extension allows you to spend an extra day in Busan, exploring the city's charms and enjoying its nightlife. Visit the dragon temple, Yonggungsa, with its rocky cliffs overlooking the sea, or the ancient Beomeosa Temple. For a bit of relaxation, take a soak in the hot springs at Heosimcheong Spa. Enjoy dinner overlooking the bridge from one of the cafes on Gwangalli Beach or take a nighttime stroll in Haeundae.
Days 9 & 10: Korea's Volcanic Island
Catch a morning flight to Jeju-do. If possible, pick up a rental car at the airport. Otherwise, take a bus to your hotel and check in before heading over to the Jungmun Resort Complex. Explore the area, including the Yeomiji Botanical Garden, Jusangjeolli Cliffs, and the Teddy Bear Museum. Visit the Jeju Folk Village, check out Manjang Cave, and either enjoy the scenery around Seongsan Ilchulbong or hike part of South Korea's highest peak, Hallasan.
Day 11: Back to the Mainland & Tranquil Haeinsa
Take a flight back to Busan or Daegu. Grab a quick lunch before catching a bus to South Korea's most celebrated temple, Haeinsa, in Gayasan. Do an overnight temple stay here or overnight in Daegu.
Day 12: Traditional Aristocratic Culture
Take a bus or train to Andong and explore the Hahoe Folk Village and the Mask Museum. If you don't mind roughing it a bit, overnight in a minbak (homestay) in the folk village and enjoy an old-fashioned Korean meal.
Days 13 & 14: A Taste of South Korea's Scenic Beauty
Take a bus to Donghae or Samcheok and spend the afternoon exploring the beaches and the caves there. Spend the night nearby and the next day take a bus to Seoraksan. Spend the day hiking the various trails, exploring the natural beauty of the area and its many waterfalls and valleys. Take a bus back to Seoul to catch your flight home.