The subways are great in Seoul if you can manage them. They are numbered from 1-8 and color coded for ease of use (though the colors often are so similar it gets confusing). You can buy per ride or buy 5000 or 10000 Won tickets. Your best bet is to buy a multi ticket, especially if you're staying more than one day. When looking at the subway map, make sure you note the transfer stations. There will be a "yin-yang" symbol on the map which means you can transfer. I missed a stop a number of times because it didn't stop where I thought it should have.
Taxis are very cheap. DO NOT take a taxi that says it's a DELUXE taxi. Normal taxis start at 1600 Won. Deluxe automatically start at 4000. You do the math. The ride is the same either way. You'll usually find "deluxe" taxis outside "deluxe" hotels or offices.
THe taxi's are really cheap here, they run off meters. Despite what people say of how they like to rip off Visiters becuase they have no clue were ther going becuase the city is so confusing. Even if you did get ripped off the fares are so cheap anyways. They do bump up the prices more at night. Also make sure your on the right side of the street before you get in the taxi. That lesson cost me the extra 2000 Won. I'm giving it to you for free.