Catch the changing of the guard ceremony just outside the front gate. 11am on Saturdays (unsure about other times). Covered under the Busan temple pass which gives you access to 5 for 10,000 Won.
Nice and peaceful park in the center. Changing guard ceremony is even better than in Gyeongbukgung. Can be visited as a part of a 5 palaces and 1 shrine ticket
Every last Wednesday of the month, the palace is open until 10PM and entrance is free! Try to go if you can, it's so beautiful at night!