Why Hwaseong Haenggung Old Palace is my favorite Korean Palace
Note that the entire fortress area is called Suwon Hwaseong, while the palace is referred to as Hwaseong Haenggung. The palace, fortress, and other historical structures/areas on the grounds have their own admission fees; I provide details on the palace only below, but combination tickets are available.
Suwon Hwaseong Haenggung opens at 9:00, closing at 17:00 in winter, 18:00 in spring and fall, and this year it's open until 21:00 for cultural activities in June through September. Admission is only 1500 won ($1.50).
Access is a little tricky but manageable. Take the subway Line 1 to Suwon Station, and use Exit 6. Cross the road, turn right, and walk about 100 meters to the Yeokjeon Market bus stop. From there, catch a 7, 7-2, or 32-1 bus to Hwaseong Haenggung.
Alternately, take bus 1007 from Jamsil Station to Hwaseong Haenggung (runs every 8-12 minutes), or bus 3000 from Gangnam Station to the Buksu-dong stop (runs about once every 20 minutes).