I lived/worked in Ulsan, Korea for a long time. Every major city is very close to many outdoor activities. Koreans love to go hiking due to the mountainous terrain. It has become a national tradition. Seriously, look at each major city's tourist website and you'll see mountain hiking as a popular thing to do.
I highly recommend Seoul or Busan. Both are great cities and major hubs to other nearby places. Plus, both allow you to experience the city and let you easily escape to nature. Public transportation is excellent, easy, and efficient all over the country. Eonyang in the western part of Ulsan (in the Yongnam Alps) is famous for its mountains, hiking, and Paraeso waterfall.