At some point, you may be subjected to the indignity of having empty cab after empty cab passing you by no matter how frantically you wave your arm. This happens because many taxi drivers are reluctant to pick up foreigners who might not be able to explain where they want to go in Korean and the taxi drivers cannot speak English. So, try not to contribute to the dumb foreigner mind frame taxi drivers often have by learning how to say where you want to go in Korean or just taking the bus.
Do not be surprised if during your little trek, the taxi driver pulls over and picks up a couple more people. While double fares are illegal in Korea, it is an extremely common and popular practice. Your fare will not decrease simply because you are sharing the cab. If you are sitting in the back and two people are wanting to get in, politeness demands that you get out and move to the front.
On the whole, I've found taxi drivers to be relatively tolerant of my presence in their cab, sometimes pleasant, sometimes shy, sometimes excited. One time, after checking to see that I was not in a rush, the taxi driver did not let me out at my stop but continued to drive me around to show me the neighborhood and practice his English. You can also take the same advantage and use them as hostage listeners while YOU practice YOUR Korean. I practiced my favorite Korean phrase on a cab driver who joked about turning my dog (who was with me) into dog soup. Joo-guh-lay means, "Do you want to die?" The cabbie about died laughing and nearly hit a bus coming at us in the next lane.
Women, you should not take a taxi alone at night. Korean women will often caution each other against it. You are much more vulnerable than they are because chances are, you can't even scream for help (in Korean) much less try to reason with the cabbie and make an escape. Also, if you don't know the route, you won't know if he deviates from it. I advise that IF you DO take a taxi after dark, then only take it if you know the route and begin protesting at the slightest variation. If he doesn't get back on the route immediately, exit the taxi at the first stop or whenever he slows down enough. Just like in your own country, you shouldn't take chances.