A fun activity to do in Seoul is to walk along the stream, Cheonggyecheon (청계천). It’s a 5.8 km (or 3.6 mile) walk. In the summer, I prefer finding a shady spot to sit in and dip my feet in the water. In the winter, I’d just walk quickly along the stream as it’s really cold outside.
I recommend starting in Cheonggye Plaza. You can get there from Gwanghwamun Station (Seoul Subway Line 5), Exit 5 or City Hall Station (Seoul Subway Line 1 & 2), Exit 4.