You will be able to find a taxi anywhere in Seoul. You can hail one in the street, but beware how you do it. Putting your hand opalm out can be regarded as rude. Taxis have advantages but also two major disadvantages. Firstly, especially when it's wet, Seoul traffic can be atrocious and your cab ride could be very very very slow.  This is also likely to be the case in the rush periods.  Seconldy taxi is the most expensive way to get around the city.  However the cost is much lower than western countries snd you don't tip.

For most people, it is advisable to use the subway for most of your transportation needs. You may be OK with the buses but-apart from airport limousine buses and other express buses- bus travel can be quite hard. Many buses will only have Korean language signs, and you also need to know the code for the type of bus -is it blue, yellow, greeen or orange? UNless you know, stick to the subway. .However if you are transporting 2 or more people and heavy luggage taxis are probably your best option.  There are some things you should keep in mind. There are three types of taxis in Seoul.

 The first type is a normal type of taxi that you will find driving all around the city. They are often cream or beige, The usual fare for the ride starts at 1900 won and then goes up depending on the distance traveled and the time it has taken. Between midnight and 4 a.m. the price will be increased by 20%. Not all of the drivers will speak any English, but most of the taxis are equipped with special free interpretation devices. If the cars do not have it, the drivers can use their own cell phones to call 017-200-3000 to get an interpretation. One useful phrase is "yoggi-so se wo juseyo", meaning please stop here".

The second type of taxi is the Deluxe Taxi, also known as Mobeom Taxis. These are black and have a yellow light on top of the car. These cabs offer high class service, offer a receipt, and do not have a late-night charge. This is a good idea if you are a businessman in a rush needing to get to the airport, but not especially necessary if you are just a regular tourist trying to get around.

 The third option is a call taxi, which is good if you are in an area far away from everything where there are no other taxis. It will take about 10-15 minutes for the taxi to arrive, and there will usually be a 1,000 won surcharge, but it is better than being stranded.

  Also it is a good idea to have the address in Korean if possible.  Some of the cabbies don't read the english version of the Korean language.

Rental cars are possible, but will cost you a lot of money. It is also hard for Americans to get used to the driving style in Korea, and it can sometimes be intimidating since lanes seem to be optional, red light running is rampant, horns are used more frequently than turn signals, and motorscooter riders are definitely suicidal.  Oh yeah, and the cost of gas is extremely expensive.