Han River or Duksoo Palace Stone wall trail would be good places for a walk.
If you want to take a walk while shopping, then I would recommend Shinchon, Hongik Univ, Myungdong, the East Gate, the South Gate, or C.O.E.X(in Kangnam).
Namdaemoon(the South Gate) has many whole sales stores, so you can think of a huge market. Myungdong has many maker shops.
Gyung Bok Palace can be divided into 8 sections. Most recommended sections for tour are Geun Jung Jun, Gyung Hwe Roo and Hyang Won Jung.