Korea has many national parks but I do not know where to start?

Korea has 22 national parks and they're well worth it. For a first visit, Bukhansan right in Seoul is the easiest to reach by subway and has trails for all levels, while Seoraksan near Sokcho is the most spectacular, especially in autumn, and has a cable car if you don't want a hard hike. Entry to the parks themselves is mostly free now, trails are very well marked with Korean and English signs, and you just need good shoes and water. Check the english.knps.or.kr site for trail maps and closures, and avoid peak autumn weekends at Seoraksan if you don't like crowds.