Why don't you visit ancient city of  Korea.  Those cities are from Joseon Dyansty(AD1392 ~ 1910), Silla Kingdom(BC57 ~ 935) and modern city.  

Andong is Confucius village of Joseon Dynasty. It is famous for great scholar Mr. Ryu who acted as major high officials in Joseon Dynasty. 

Gyeongju had been capital of Silla Kingdom for about 1,000 years. It has so many national treasures. Namsan is called wall-less museum. 

Busan is also old city. It was only remain during Korean war. Nowadays it is biggest harbor city in South Korea with fresh fishes. 

Mt. Halla in Jeju, S. KOREA is one of the best mountains for snow trekking. It is 1950 meters high. Normally you will start 800 meters high sea level. There are two courses to hit the summit, Baekrokdam. Those are Seongpanak course and Gwaneumsa course. Other three courses are not hiting the summit but those are still beautiful. For beginners start from Seongpanak - Summit - return to the same Seongpanak is highly recommended. These are 9 hours round trip hiking courses. So you have to bring lunch in your backpack. For your safety, there is time limit. By 12:00pm (at noon) you should reach Jindalre shelter from Seongpanak (or Samgakbong shelter from Gwaneumsa point). I recommend you to reach Jindalre shelter by 11:00 am. Because you need to have a lunch. At least you have to wear crampons (=eisen) and spats. Sticks also helpful. Prepare one time use raincoat for warmer as well as rain. Among them crampons are necessary. Bring hot water and drinking water. Have a nice trip. If you want guide, please book me here in Synotrip. Please feel free to ask me for questions. Kevin Dream.