In the heart of the city, Gyeoungbokgung Palace is the largest and most famous of the five Royal Joseon Palaces. You can borrow hanbok around the palace (multiple stores nearby. price is rangin $10-100 USD), and you can get in to the palace for free if you wear hanbok. If you have more time or love visiting palaces, I'd also recommend Changdeokgung palace, which is a UNESCO World Heritage Site and home to the Secret Garden (Huwon).