The route is simple. The easy way is to take the subway, and according to an online calculator, it takes 30mins and 1250Won and you feel lethargic and unhealthy after the trip. Let me show you another way, about 6 hours, free apart from some bananas, powerbars and 2 litres of water and plenty of patience. In return you will get some nice sceneries and fresh air. With all the nice mountains around Seoul, there's no excuse not to do it the old fashion way.
Other more reputable sites will tell you what is in Seoul Land. I've only been to the zoo and only because it was free that day. Will it be a surprise if I said there are animals there? In confined spaces. With plenty of noisy kids. Discovering that there are no birds to fling on catapults. So collect yourself at the carpark, run up Cheonggyesan 청계산 due east, come down where Dallaenae-ro 달래내로 meets the expressway, and then crossing some shops and farms, proceed up the second mountain Inneungsan 인릉산 and end up somewhere in Seongnam, close to the airbase and then run along the river up north to Jamsil.
For this trip, I took a backpack with some emergency kit and wore a Suunto Ambit. The screenshot above is what was recorded for the trip. The plots are quite rough but it should be able to hack together a plan for the trip. I find that the best map to use if you had a phone with you would be Daum or Naver since they have the map of the trails. Be aware that the maps are not entirely accurate, sometimes there are sub-path along the way that was created recently. Sometimes they are parallel paths and sometimes it looks like a whole mob went for a shit together and created a new path.