Busan, South Korea is an amazing city. I liken it to San Francisco because of its mountains and the seaboard. It is extremely modern, has impeccable public transportation, and is a booming metropolis being the largest port in South Korea.
It is probably a fortunate thing that the city is so mountainous and those mountains so steep that the urban sprawl was contained, and as a result, one is never far from the havens of the mountains of Busan.
The hiking trails are abundant and many of the city's most beautiful temples can be best accessed by hiking through the mountains.
Hiking is extremely popular with the aging population of Busan and there are several high-end hiking apparel lines that have emerged from the hiking culture here (e.g. Treksta). Koreans adorn themselves in bright, matching hiking outfits and are decked out from head-to-toe in gear. One can't help but notice their complete aversion to sun exposure and that they will wear jackets and pants even in the heat of the summer.
Another interesting thing is the presence of mountain-top restaurants that sell pajeon, soup, makgeolli, and soju. Alcohol is never far in South Korea even up in the mountains, but just remember that if you partake at the top of the mountain you still have to hike down (speaking from experience here)!
Be warned that even though the trails of Busan are usually well marked there is almost never any English displayed, so be prepared to ask for directions! A metro stop is always nearby.