Also at the park is a full-on water park and a "culture expo" of some sort; I haven't checked out either.
Park hours can vary greatly depending on the season; in July they ranged from 9:50-18:15 to 8:50-20:45. Admission is 20,000 won ($20) and a free pass is 35,000 won ($35) for adults.
Google Maps has good bus information for Gyeongju (and most of Korea) so routing from Seoul can be done through its directions feature. It's also possible to take an intercity bus to Gyeongju and skip some of the steps above; ask at the visitor's center for whatever city you're in.
There is a visitor's center with minor English ability (enough to get directions and such) at the Gyeongju intercity bus terminal. The bus terminal clerks don't really speak English. As always in Korea, I recommend having your destination's name and address written in Korean characters to point at when asking directions or taking taxis.
There's also a teddy bear museum at Seoul Tower, and a few historically significant points in the vicinity.
To get to the tower at the top of Namsan Mountain, you can take a shuttle bus from Myeong-dong Station exit 3, Chungmuro Station exit 2, Seoul Station exit 9, Itaewon Station exit 4, or Hangangjin Station exit 2. It costs 950 won in cash, but it's cheaper if you use a transportation card and transfers are available.
From Exit 3 of Myeong-dong Station, you can also walk a ways to a cable car (8500 won round trip, 6000 won one way); this route includes using a free slanted elevator that is pretty slow. The shuttle buses are cheaper and faster. If you really want to walk up/down the mountain, it looks like the shortest route is from Seoul Station, though finding good details online is pretty tough.
North Seoul Tower is open daily, 10:00-23:00 (10:00-24:00 Fridays and Saturdays). The six restaurants open by 11:00 and close at 23:00 (22:00 for the snack bar). The observatory costs 9000 won ($9) for adults.
Take subway Line 4 or Jungang Line to Inchon Station and use Exit 2. There is an underpass from the subway to the museum called "Moving Museum" that was a nice addition.