The public transport network is fantastic. Our general experience spread over our four year sojurn was that the buses and trains ran on time and were very safe to travel on in all extremes of Korean weather and taxis were readily available if required. Depending on which town or city you end up teaching in, you will be able to access at least two modes of public transport, or if you are very lucky all 3.
In our town of Buyeo, we only had access to buses and taxis. This was due to the fact that Buyeo is a town with great historical significance and therefore the train network is prohibited. The nearest train station to us was around 40 mins away in a town called Nonsan.
If you find that you are in a town where there is no train, your only way to travel around the country is by using the intercity bus network. Although slower than the train, we made extensive use of this network, travelling around and visiting different parts of the country.
If you do prefer train travel, then making use of the Korea Railroad Corporation promoted as Korail (코레일) is your best option. Two most popular types of rail transport in Korea are the Korea Train Express (KTX) or the slower Mugunghwa trains. The KTX is quite expensive but if you have the need for speed, or need a time saving commute, then the KTX is the way to go.
Taxis as previously mentioned are readily available, but can be expensive. Although cheaper than many countries, the bill soon adds up. It is best to use the local town buses to get around.
Although having said that, taxis are very convenient, especially after you have been shopping or its raining cats and dogs.
Taxi drivers can also display unfriendly attitudes towards foreigners. We experienced very few instances of this, as the majority were very helpful and friendly. Just be aware that this may be the case.
We often thought about purchasing a car (many foreigners do) but considering the financial outlay and insurance we would have to pay on the vehicle we decided against the idea.
Another reason we encountered which factored into our decision against purchasing a vehicle was what if we had an accident and had no easy way to communicate with the police? We had heard rumours that the traffic police in the case of accidents generally would side with the Korean driver-whether there was any truth in this or not we decided in the end public transport was the best and safest option.