You can see the cherry blossom during 4th week of March and and 2nd week of April.
Cherry flowers come into blossom as the temperature reaches around 15C degrees. Therefore, cherry blossom is moving to north from JEJU ISLAND.
Whereas, the autumn color moves on the contrary.
Maple, or gingko trees turn yellow/red around 10C degrees from north-east mountain.
Therefore, you need to go to MT. SEORAKSAN around 2-3rd week of October. If you stay at SEOUL, you can see some gingko trees with yellow leaves only. (I.e., you cannot see the of mountains covered with yellow/gold, or red color. )
But, if you cannot make time in October, you can go to the south-west of KOREA. There are some famous mountain for autumn color such as MT. NAEJANGSAN, MT. JIRISAN. I.e., The peak timing at these mountain is 1-2nd week of November.