There are few zoo in Korea, but the nearest to Seoul is the one is Seoul Grand Park.
Various animal from around the world is there, and the place is really huge.
You can even find endangered species being placed there with care.

You need to take a whole day (say from Morning to evening) to visit the entire zoo.
I would strongly suggest sport shoe or flat, as you will need to do lots of walking.
No need to pack food, as there's lots of it inside with reasonable price.

Another thing to note, there's animal show too.
Make sure you come early in the morning or check the website to know more in details.

Website : http://grandpark.seoul.go.kr/Eng/html/main/main.jsp

Once you exit the subway, you'll see a huge building at the back of the fountain.
From there, you can buy tickets to take mini train to enter any of the park
(There are Seoul Zoo, Rose Garden, Theme Park, Children Park all in one big area)
 

Since we will spend one whole day, there, I suggest you not to be too greedy, take one by one first.
Plus, Seoul Grand Park, always have upcoming interesting new event.

Back to the entrance, before you enter, there's 2 counter on the left side, that offer package option.
If you would like a package train + entrance + sky lift, then I would suggest you buy it there.
If not, you can just buy mini train ticket before you enter.

(Please note that my pictures are from different season)