I was in Korea same time as Joel but took the opposite route starting from Gyeongju to Seoul. I had been to Seoul twice before. In 2011, I was there for 6 weeks studying and partying, did not do touristy stuff but became an expert on the city's nightlife. In 2015, i stayed for a week but managed to visit only Jongmyo Shrine and parts of the City Wall. This trip, I am backpacking and focused on seeing WHS. I was joined by my parents in Seoul and visited Changdeokgung with them.
As Joel said, Google maps is not fully functional in South Korea. However Google Maps is reliable for the public transport options. It shows you which bus/train to take, what time they leave and where are the stops. Google maps DOES NOT show the GPS tracking while you travel. And that is where maps.me (offline map) comes in handy. Outside of Seoul most bus/trains and some other public areas have Public WiFi. Some new buses have public WiFi too. I was able to skilfully track and plan my routes using these public places, google maps and maps.me. (I was in roaming but since my SIM is based in China, I need to use a VPN to access google maps which makes the connection slower).
Day 1: (Heinsa, Jisandong Gaya Tumuli, Gyeongju Tumuli Park) Overnight in Gyeongju
Arrival in Daegu at 0530. Took bus to Heinsa at 0720 (First bus is on 0640 and leaves every 40 mins). The bus passed by Jisandong Gaya Tumuli at Gyeoryo. I arrived in Heinsa before the crowd which made me appreciate the site even better. Gyeoryo is between Heinsa and Daegu (an hour each way). You may make a side trip to Dodongseowon too from Gyeoryo. I got back to Daegu at 1500 and took express bus to Gyeongju. I stayed in airbnb near the Tumuli Park and the Observatory and was able to see it at sundown and early evening.
Day 2: (Oksanseowon, Yangdong, Bulguksa, Seokguram, Gyeongju Wolsong Belt), Overnight in Gyeongju
Took the 0730 bus to Oksanseowon. Arrived at the site by 0815, free entry. The bus back to Yangdong is at 0840, which I missed and I had to wait for the next bus at 1020 (wasting almost 2 hours). 20 mins at Oksanseowon is more than enough so you should try to catch the 0840 bus. Arrived in Yangdong before 1100 and I only stayed 2 hours in the area and managed to see 3 of the 4 national treasures. Took the 1250 bus back to Gyeongju and took bus to Bulguksa. Stayed in Bulguksa for around 2 hours then too bus to Seokguram where I stayed only less than 1 hour as nit much to see (the grotto is impressive though but you cant get close or take photo). Bus back to Gyeongju visiting Anapji Pond and Walking at Wolsong Belt.
Day 3: (Tongdosa Temple, Baekje Songsanri), overnight in Gyeongju
Took 1000 bus from Gyeongju to Ulsan then transfer to local minibus to Tongdosa. Tongdosa Temple is at the opposit end of the town, around 30 minutes walk from the bus station. Stayed in Tongdosa for 2 hours, walk back to the station and decided to take a can to Ulsan KTX. KTX train trip to Daejeon took around hours. Bus from Daejeon to Gongju. Arrived in Gongju at 1600. Visited Songsan-ri Tomb and Museum until close.
Day 4: (Baekje Gongsanseong Fortress, Magoksa Temple, Dosanseowon)
Took 0800 morning stroll at Gongsanseong. I was lucky that the site was already opened an hour early and free. Usual opening time is 0900. After walking around the fortress, I took a taxi to Magoksa (12km away from Gongju). Stayed in for 2 hours then took bus by 1300 back to Gongju. Transfer to bus from Nonsan-si (less than an hour ride) and took local bus to Dosanseowon. Strolled around the Seowon for an hour then bus back to Nonsan Train Station to Suwon (2 hours away). Arrived in Suwon early evening.
Day 5: (Namhansanseong, Illeung Royal Tomb, Hwaseong), overnight in Seowon.
Travel to Namhansanseong around 0730. Arrived at the South Gate around 0900. Walk around the vicinity for an 30 mins (there is nothing to see, it was raining and cold). Walked to the Emergency Palace. After visiting the palace, had lunch in the area and tracel to Illeung Tomb by bus (1 transfer). Had the sight to myself and i walked around both tombs in the area. Then took a bus back to Suwon to Hwaseong. Walked from the entrance of the fortress till the end at the otherside near to Suwon downtown.
Day 6: (Yungneung-Geolleung Tombs, Gwanghwa Dolmen), overnight in Seoul
Arrived at the Tombs by 0900, walked around it then travelled to Gwanghwa by series of trains and buses. Due to heavy traffic from Swine Flu checkpoints, areived in the Dolmen around 1500. Only had 20 minutes to see the largest Dolmen area (there is nothing to see anyway). Then bus to Gimpo Airport to meet my parents. If there was no traffic jam, I doable to squeeze the Royal Tombs at Gimpo. (The bus from Gwanghwa back to Seoul will pass by the area. Stop at Gimpo City Hall and the site is 1.2 km from bus stop)
Day 7: Seoul. Free day. DMZ tours were cancelled due to Swine Flu. Managed to see the Royal Tomb near Gangnam
Day 8. Seoul. Chengdeokgung Palace.
Day 9. See Uireung Tomb in the afternoon. Fly to Jeju in the evening.
Day 10. Private 10-hour Jeju Island Tour with stops at Hallasan 1km viewpoint, Sunrise Peak and Manjanggul Lava Tubes. Also some have sightseeing other sites.
Day 11. Flight back to Beijing
PS. I dont usually eat lunch when I travel as I dont like to use public restrooms. Hence I was able to optimize my travel. All in all, mainland South Korea can all be seen in 7 days.