Getting Around Seoul
Traffic laws, as well as mere courtesy on the road, seem optional. Be extremely cautious crossing streets, and take liberal advantage of underground walkways, and use subway entrances as a quick, safe way to cross streets.
Even on the sidewalks, keep the kids nearby. Motorcyclists will often drive up on the sidewalk to get around traffic, and can come up on you quite suddenly. Never assume that as a pedestrian you have the right of way, and at intersections watch the cars themselves rather than relying solely on traffic lights or WALK/DON'T WALK signs. Even older kids will need some time to adjust to the new rules of the road before being able to cross streets alone safely.
The good news is that the subway system is clean, safe and goes everywhere. You can buy your ticket either from a machine or at a window near the entrance to the subway. “Kids” go for half price; I am not sure what the official rules are, but our nine year old always has to pay while our five year old usually does not. Once you pass through the wicket, keep your ticket as you will need it to exit the subway at the other end. We always hold on to the kids' ticket(s) ourselves to keep them from getting lost enroute, though they find it fun to stick their own ticket into the machine at our destination.
Taxis are available easily, at least during the day and when it is not raining. You can hail one on the street. Black taxis are the deluxe models and charge more for a bigger back seat and air conditioning. The grey taxis are slightly smaller and cheaper to ride. It is best to have you destination written in Korean if you cannot speak Korean.