It is a bit tight to do this in two weeks, but it can be done. In fact Lonely Planet’s two-week itinerary somehow packs in a few more stops. Buses are frequent and seats do not normally need to be reserved.
If short on time, you could spend less time in Seoul and/or cut out either Chuncheon (and get a bus straight to Sockho from Seoul – 2 hours, 40 mins) or Busan (and get the KTX train from Singyeongju back to Seoul – 2 ½ hours). Gyeong-ju and Gangneung and surroundings could easily occupy you for another day or two.
Day 1-5 – Seoul
Do: Palaces (go as early as possible, before the crowds); Buckhon hanok (old wooden houses) village; modernist architecture; day trip to the DMZ (note that the JSA is not open on Sundays); walk along the Cheong-gye-cheon inner city stream; hire a bike in Yeouido Park (Yeoui-Naru metro station) and cycle along the Han River; hike Namsan; Seodaemun prison.
Eat: surasang (royal cuisine); bulgogi (marinated beef)
Day 6 – Chuncheon
Get there: morning Intercity bus from Dong-Seoul (1 hour, 10 mins).
Stay: Zak’s Guest House (http://zaksguesthouse.com/)
Do: Hire a bike (ID needed) and cycle round Uiam Lake.
Eat: Dakgalbi – spicy chicken pieces
Day 7- Sokcho
Get there: morning Intercity bus (2 hours)
Stay: Mr Egg
Do: Take local bus 7 or 7-1 to Seoraksan National Park (numerous hikes of anywhere between 1 and 7 hours); fishing port and fish market
Eat: Crab; modeumhoe (mixed raw fish); Sokcho stuffed squid
Day 8-9 – Gangneung
Get there: morning Intercity bus (1 hour)
Stay: Aark House
Do: Take shuttle bus to Jeongdongjin for Unification Park (captured North Korean spy submarine); walk or cycle round Gyeongpu Lake; Gyeongpu Beach; Seongypjamng and Ojukheon (stunning historic complexes, within walking distance of each other).
Eat: Fish; seafood (at Gyeongpu)
Day 10-12 – Gyeong-ju
Get there: morning Intercity bus, via Pohang (c.5 hours)
Stay: Coolzaam Guesthouse (http://www.coolzaam.com/)
Do: royal tombs; temples; cycle/hike in several attractive parks; folk villages; 7th century observatory; Namsan hikes – from 3 to 8 hours
Day 13 – Busan
Get there: Morning Intercity bus (1 hour), or local bus to Singyeongju and KTX train (30 minutes)
Stay: Angel Hotel (http://www.angelhotel.co.kr/)
Do: Jagalchi market (Korea’s equivalent to Tsukiji in Tokyo); Haeundae Beach
Eat: Hoe (raw fish) or choose your own fish at Jagalchi and go eat it upstairs.
Day 14 – Seoul*
Get there: KTX train (3 hours)
*Alternatively, take the Beetle Ferry to Fukuoka in Japan and carry on travelling