Two weeks in South Korea
Established since 1945, South Korea is the only country that is still divided. Many of its historical sites were damaged during the Japanese Colonial Period and the Korean War, but Korea still remains an interesting place to visit. As the Koreans would say, the country is five millennia old, and the achievements don't just disappear overnight.
This itinerary is made for those who want to see and experience Korean culture during their travel. It covers various cultural sites in South Korea, including both traditional and modern.
Learning Hangeul, Korean script, is quite essential since many signs are written only in Korean. English is taught widely, but you can't expect everyone to speak English due to the lack of practice.
 Get in/out
Most travelers enter and leave South Korea via Incheon International Airport. Entering South Korea via North Korea (and vice versa) is impossible unless you want to risk being shot. There are some ferries from Japan and China.
Incheon (Day 1)
If you arrive early in the morning, you may take an airport bus and head to Seoul right away. If you come after a long flight from Europe or North America, find a hotel in Incheon and relax. If you're of Chinese origin, you'll find it worth to visit the only Chinatown in Incheon. It's the only Chinatown in Korea, and it's located nearby Incheon Station. Visit Weolmido (Weolmi Island) if you have enough time.
Seoul (Day 2-4)
After a good night of sleep, take the Metropolitan Metro in Incheon Station. It takes a about 3 hours to get to the downtown of Seoul, and it costs only about USD 2. You won't get there any faster in the morning, the traffic jam is very serious and thus taking a taxi will cost quite a lot of money.
There are four palaces in Seoul, and it'll take at least 2 days to see them all. Spend a day seeing Palace Deoksu and Palace Gyeongbok (City Hall Station), and another day seeing Palace Changgyeong, Palace Changdeok, and Jongmyo, the Royal Cemetery (Jongro 3-ga Station). There is a folklore museum in Palace Gyeongbok as well. The order has a meaning. Palace Deoksu and Palace Changgyeong have been damaged much in the 20th century, so if you visit them in reverse order, you'll find Palace Deoksu and Palace Changgyeong unimpressive.
In the third day, visit Insadong (Jonggak Station), where numerous art galleries and traditional souvenirs are available, and Itaewon Street (Itaewon Station), the most cosmopolitan part of Korea.
If you want to enjoy nightlife, Seoul is a perfect place to do so. It's even safe to walk around in 3 AM all by yourself. Take a metro and stop at Hongdae Ipgu, Shinchon or Hyehwa, you'll be dazzled by neon signs. In case you're lost in Seoul (or anywhere in the country) there will be a convenience store on almost every block, and they open 24 hours. Find one, buy a small thing ask your way.
DMZ (Demilitarized Zone) - Panmunjeom (Day 5)
Go to Seoul Station early in the morning and get a ticket to Dorasan. You'll have to stop at Imjingang Station first, and then reserve a guided tour of Dorasan there. DMZ is a controlled area and you can't travel there alone, and it is not safe to do so, due to the land mines. Imjingang Station used to be the northern-most station in Gyeongeui Line, and there you'll find some memorial sites of Korean War and an amusement park. Look around the places; you'll have to wait a few hours until your train leaves to Dorasan.
In Dorasan Station, you'll find the soldiers doing all the ticketing, but don't be afraid of them, they're very kind to travelers. You will have a guided tour around the place including a North Korean tunnel called "Third Tunnel" and Dorasan Observatory. You'll be back to Seoul by evening.
Gangneung (Day 6)
Head to Cheongryangri Station and take a night train to Jeongdongjin. Make sure you get there before the sunrise; Jeongdongjin is known for its beautiful sunrise. Have a breakfast and head to Gangneung, just north of Jeongdongjin. Visit Gyeongpodae beach, one of the best in South Korea. It's easily accessible by bus from downtown Gangneung. But don't go there in early August, you'll find it stressful because it'll be as crowd as Seoul Metro in the morning. Treat yourself with some fresh sashimi and relax.
Andong (Day 7)
Take a train from Gangneung and head to Andong in the morning. Andong is one of the most traditional places in South Korea. Visit Dosan Seoweon (a Confucian Academy) and Hahoe Village to see some traditional places. Here, you're better off to take a taxi, public transport is not as good as other places.
Gyeongju (Day 8-9)
The traditional stuff you've seen are all medieval Joseon style, but Gyeongju is different. It's been the capital of Silla, the kingdom millennia. Visit Cheonmachong (royal tomb), Bulguksa (Buddhist temple), Seokgulam (temple in the cave), and Poseokjeong (a ceremonial site). It will take at least two days to see Gyeongju.
Gwangju (Day 10)
In order to go to Gwangju, it'll be better to take bus than train. Here, the entire city is a historical site. Gwangju has been most rebellious city in 20th century history. Two uprisings led by students - one against Japan, the other against military dictators of South Korea - took places here. Geumnam Street in downtown was the battle site between civilian army and the government army in May 1980. Visit the ceremonial cemetery in Mangweoldong, it's a symbol of Korean democracy. Part of the bodies were moved to May 18th National Cemetery in the outskirts of the city, but the one in Mangweoldong is more symbolic.
Buyeo (Day 11)
Take a bus early in the morning and head to Buyeo, the last capital of Kingdom of Baekje. The kingdom fell in 660 and the city was razed, so you'll see no Baekje castle here. The castle in Buyeo is rebuilt in Joseon era, but it's still worth a visit. Then visit Buyeo National Museum, and see the remainder of Baekje art. Many Koreans find Baekje art more elegant and subtle than Silla.
Gongju (Day 12)
Take a bus to Gongju, the second capital of Baekje. It takes less than an hour from Buyeo. Visit Tomb of King Muryeong. You won't be able to enter the actual tomb, but they have a good replica. Then visit Gongju National Museum. It's quite hard to find the place due to the insufficient road signs, so you better call a taxi.
Suwon (Day 13)
Take a bus from Gongju to Suwon (about an hour and a half, but make sure you don't get off when the bus stops in Osan on the way) and see Castle of Suwon, or Suwon Hwaseong. It is one of the most well-preserved castles in South Korea, and it'll certainly dwarf the one you saw in Buyeo. From Suwon, you can simply take metro to come back to Seoul.
 Stay safe
South Korea is, in fact, a very safe place to travel by yourself. The security level is quite high, and even if you are a female traveler wandering around streets of Seoul in 3 AM, the chance is very little that you'll find yourself in danger. And if you look "foreign" (especially White or Black) the chance is higher that Koreans would be more afraid of you than you are. (They're just not used to the people other than their own.) Still, a common sense applies here.