Yes, I would definitely recommend that you visit Changdeokgung first. To me, it's perhaps the most beautiful palace in Seoul. And yes, there's a fee to get in. It costs 3,000 Won just for a basic tour, and you can only visit the palace on a tour that you can sign up for at the admission office. I would also recommend paying the 5,000 Won extra. And if you're thinking about visiting all the major palaces in Seoul like Gyeongbokgung, Deoksugung, etc., I would recommend getting the integrated pass for 10,000 won.
Hope that helps.