There are two major amusement parks in Seoul, and really all of Korea. The first one I visited was Everland, a sprawling park in the mountains south of Seoul itself.
A one-day ticket is 46,000 won (about $46), while an afternoon ticket (after 16:30) is 38,000 won. However, foreigners should visit Everland's website and print the coupon to get a full day admission for 37,000 won. The park is usually open about 10:00-22:00.
There's an "Ever Line" subway/train that goes to the park's transportation center, but for most people it's much more expensive and takes a lot longer than riding a bus from Gangnam or Jamsil. The public bus (2000 won) is more than good enough, though you can pay about six times the fee to ride a reserved bus (12,000 won) that departs from several locations around town once in the morning, returning in the evening.
Seriously, take the public bus from Gangnam or Jamsil. The reserved bus will put you on a specific schedule that cuts your day short, and the subway is slower and more expensive for most.
Lotte World is located in the heart of Seoul, but my feeling is that Everland is a better park. Granted, I've had several problems with Lotte World's crowds and ride closures (which I'll detail in a Lotte World post in the future). Note that like everything else fun, the park will be very busy on holidays and possibly later in the afternoon as people take advantage of evening admission prices.