Transportation in Korea comes in many many varieties and is usually of exceptional convenience and reasonable cost. 

Buses

Buses thoroughly cover any city or town you are in, and during regular hours they arrive every 5-15 minutes, going to just about anywhere you want. Taking a bus is probably not something you wouldn't want to do until you are reasonably comfortable with the areas you are going to. Reading the list of districts on the side of the bus is quite difficult to do when the bus is whizzing past you at full speed ...oh, and it's usually written in Korean, so brush up on those reading skills. Over the past few years, they've been adding more and more English to the busses and other modes of transportation in Korea, so depending on where you are, you might get lucky! If you know the name of the area you are going to (and can pronounce it with the proper Korean intonation), you can usually say that name with a questioning intonation and the bus driver will nod yes, or he'll motion for you to get off the bus. If he nods, and you get on the bus, don't count on him to tell you when it's your stop. Some will, some won't

Flagging a bus down

Most bus stops have numerous buses that come through, so the bus won't always stop if you're just standing there waiting. You'll have to wave it down, or it may just pass you right by. To wave one down, put your arm straight out, and with your palm facing down, move your hand in an up-and-down motion. Just putting up your hand or waving frantically at a bus won't always make it stop. It's actually considered somewhat rude to do so in Korea, and whistling is a definite no-no!

Getting on and off a bus

Once you are on the bus, hang on for your life! I mean, from the moment you step foot on it (two hands, if possible). Busses are fast and they own the roads, which often makes taking the bus the quickest option. If you're on one and you need it to stop, there are little red buttons placed all over inside the bus (even on the ceiling). Press one of those before your stop rapidly arrives. Most buses in Korea have a notification system that announces each stop as it nears, followed by the stop after that. They will announce subway station stops and some tourist areas in English. When you're standing and waiting for the bus to stop, again, hang on and make sure you have your feet positioned for an abrupt stop.

On any public transportation, you should always give up your seat (if you even have one) for the elderly. I'm sure it's the same where you're from.

Fares

The cost of riding a bus depends on the type of bus it is and where you are. It is anywere from 550 won to 1500 won, and it is usually clearly posted on the change meter next to the driver. While waiting, you can ask someone near you "Buseu, all ma yay yo?", meaning "How much is the bus?"

The yellow city buses cost a flat 1100 - 1200 won

The smaller blue buses cost about 600 won and only circuit the immediate local area

The larger burgundy or white express buses are about 1300 won and are often much more convenient to get across town but they are less frequent.

Hours of Operation

Buses in most areas will run until about 11:30 pm and start up again at around 5 am. There is sometimes a sign posted with the schedule on it but it's rarely accurate. You can usually trust that one will arrive shortly.

Limosine Buses

First of all, from the airport, DON'T take a taxi. They'll try to charge you anywhere from 90,000 to 200,000 won flat-rated for a trip that should cost about 45,000 won. From the airport, you can take a limosine bus right outside the terminal to any part of the city. They arrive every 20-30 minutes and cost from 7,000 to 15,000 won, depending on where you are going. The information desk or limosine bus desk can guide you to the correct bus. Limosine buses are only 3 seats per row and very comfortable!

You can also take limo buses from express bus terminals to pretty much any part of the country! The cost is a little more than regular bus service but it's well worth it! A trip from Seoul to Busan (5 or 6 hours) costs about 35,000 to 40,000 won.

Buses in Seoul

In Seoul, since 2004, the busses have been color-coded and the bus routes simplified. They are now running through specific districts which are correspondent to the bus number.

Usually, when a bus has the name of the district on the front or side of it, the names will have "-gu" or "-si" added to the end of them.

The suffix "-gu" means "ward" for the numbered colored areas on the map, within Seoul. There are a total of 8 wards/districts within Seoul.

The suffix "-si" means "city" for the numbered white areas on the map, which are satellite cities. There are a total of 7 cities that the Seoul suburban bus routes run service to.

Types of Buses (color system) and Fares

The blue buses are long distance buses and serve more than 2 districts. They have a 3-digit number on them that stands for Departure District + Destination District + Bus ID number (0~9)

(Example) 048
0 - This bus starts from District Area 0 (Jongno, Junggu, Yongsan)
4 - This bus heads for District Area 4 (Seocho, Gangnam), with stops along the way
8 - Bus ID number

Price - 900 won

The green buses are single district buses that carry passengers to transfer points like subway stations and major intersections. They have a 4-digit number on them that stands for Departure + Destination + Bus ID Number (11~99)

Price - 900 won

Red buses are express buses that carry passengers to suburban areas. They are fast and very comfortable. They also have a 4-digit number on them that stands for Bus Type (Suburban bus numbers begin with 9) + Departure District + Bus ID Number (00~99)

Price - 1500 won

Yellow buses travel a closed circle route within major district areas of Seoul. They have a 2-digit number that stands for District + Bus ID Number (1~9)

Price - 550 won

Subway

The subway is quite a convenient way to travel in Korea. Seoul, Incheon, Busan, Daegu, Daejeon, and Gwangju all have subway lines, and some are more intricate than others. Seoul, for example has a massive web of 8 lines, with an additional 4 satellite city lines that interconnect with it (including one that goes to the Incheon International Airport). Busan has 3 lines with another line interconnecting with the city of Yangsan. Daegu has 3 lines. All other cities have 1 line with additional lines planned.

Subway signs are mostly written in Korean, English and Chinese, making it fairly easy to find your way around them.

Subway Map

A large map of the subway can be found right above the ticket vendor. Another large map can usually be found close by if there are ticket machines in the station. These maps label each stop in both Korean and English. Small tourist maps or wallet sized maps can also be obtained through most ticket vendors. To ask a vendor for a map just say, "Map, jusaeyo."

Seoul -

http://www.smrt.co.kr/Eng/Subway/Howto/Subway_method.jsp#2 

Busan - http://www.subway.busan.kr/english/03/01_01.php

Daegu - http://www.daegusubway.co.kr/source/eng/

Incheon - http://www.irtc.co.kr/eng/map/incheon.asp

Daejeon - http://www.urbanrail.net/as/daej/daejeon.htm

Gwangju - http://www.subway.gwangju.kr/eng/0301000000.htm

Fares

The cost of traveling by subway starts at about 1000 won (varies a little by city), and goes up depending on the distance you are traveling. The fares are clearly posted on the large subway maps in the stations.

Purchasing a Ticket

You can purchase a ticket through the ticket vendor or a machine (when available). If you know the price you need to pay, you can just give the vendor the money and he'll give you a ticket for the amount of money you gave him. If you're not sure, you can tell the vendor the stop you wish to travel to and he will tell you how much...usually in Korean, so knowledge of money units in Korean language would be very helpful at this stage.

Getting on the Train

You'll use your ticket to go through the turnstiles by inserting it in the ticket slot, walking throught the turnstile and retrieving the ticket wherever it happens to come out (usually on top at the other side). KEEP THE TICKET as you'll need it to exit the turnstiles at your destination station.

As each line will be traveling in two opposite directions, you'll want to make sure you get on the train going in the direction you wish to travel (trust me, we've all went the wrong way at least once). Signs will be posted overhead with 2 or 3 major stations listed on each. Knowing if one of these stops is in your direction can be very helpful (check the map). Once you get to the platform, there are additional route maps so you can double check that you're going the right direction. Across from the platform, you'll also find a route map on the wall that tells you the amount of time it will take you to get to your destination.

Usually the train will announce each stop before it arrives. The automated system will announce it in both Korean and English, but if the driver announces it....well, I think you know. Listen very carefully as the announcement is sometimes difficult to hear. The reason for this could be that the volume is low or the train is full of people (many of whom have a cell phone attached to their ear). You can also look out the window to the platform and there will be signs stating the station you are at, with the previous and next stations on either side of it with arrows.

You'll find that the seats on the subways are almost always occupied. If you do spot an open seat, you gotta move fast or someone else will take it! There are elderly and handicapped seating areas on every train, but when it's really full, it's polite to give up yours for someone who could use it more (Or....you can just put your head down a little and pretend you're sleeping until someone else moves for them).

Exiting and Transferring

As you exit the train, there will be numbered exits listed next to a yellow sign that says "Way Out". There will be a map of the area on the wall somewhere nearby that you can use as a reference if you know any of the landmarks listed on it.

If you need to transfer to another line, as you step off the train, you'll see a colored line on the wall, with the color and number of the subway line that corresponds to it (or many, if there are a lot of lines connected to that station). Follow the arrows to the new platform of the line you wish to transfer to.

Taxis

Korea is like New York when it comes to taxis. If you need a taxi, there's one in any direction you turn. Taxis stands are used in some busy areas, or you can hail a taxi from the street. Taxis are everywhere and quite cheap to use. The fare starts at 1800-1900 won for the first 2 kilometers and goes up by 100 won for each additional 144 meters or 35 seconds.

Types of Taxis

There are two common types of taxis. The regular taxis are usually silver with a white sign on top. These are the cheaper and more common types. Then there are the deluxe taxis or "mobeum" taxis. They are black with a yellow sign on top and say "Deluxe Taxi" on the side of the car. These cost 4500 won for the first 3 kilometers and an additional 200 won for each additional 164 meters or 39 seconds.

Taxi vans can also be ordered for large groups or luggage

There are also brand taxis now, called "Kind Call Taxis" that are equipped with interpretation machines and receipt machines and have a 24 hour call center. The fare is the same as a regular taxi with an additional 1000 won charge for call usage.

Taxi drivers in Korea can be quite entertaining at times. If you take a taxi often enough, you'll likely run into the Karaoke taxi, where the driver has a small speaker with a microphone, and he sings Korean traditional songs to you while he drives. He'll even let you give it a go!

Tipping Drivers

Tipping is North American culture and is generally not something you do in Korea. Tipping a taxi driver can sometimes be taken as a rude gesture, presuming taxi drivers don't make enough money. If there will be change from your fare (at least 400-500 won), it is sometimes ok to leave it behind and say something like "Copy" (coffee). Careful not to confuse this with "Koh pee" (nosebleed).

If you are taking a long ride home, very late at night, tipping a driver is a little more welcomed. Late at night, there aren't as many people taking taxis, so one could assume that if you are traveling quite far from the center of town where the party is, that the taxi driver will be returning to that area without a fare. A few thousand won is plenty enough and should only be given at your discretion, or if the driver was exceptionally kind or entertaining.

Again, it is never expected of you to tip anywhere in Korea (except maybe a luxury hotel), and in doing so can sometimes make Koreans very uncomfortable.