Transport in Korea can be surprisingly easy but also frustratingly difficult. Here are a few tips I've picked up so far.
Korean Air is the larger of the two Korean airlines. They have a good website where you can check their flight schedule and seat availability but when it comes to booking a seat and getting a ticket, unless you have a Korean credit card forget it.
The site does claim to handle certain North American credit cards, but for those of us with European credit cards you have to phone them, tell them all the details you've just worked out on their site and you get a booking reference. You have five days to go to their office or a travel agent to pay for the ticket in person. Quite why the website insists on reservation and booking in one step, rather than allowing online reservation and subsequent payment in person is beyond me.
For our Visa run we've chosen to fly to Tokyo for a few days. We will return next week on tourist visas. We had a reservation for Korean Air tickets for approx 500,000 each. However we subsequently found cheaper tickets through a travel agent GoNSee flying with NorthWestern Airlines for only 390,000 won. I recommend GoNSee for their friendly and efficient service.
The other airline is Asiana Airlines. I have no experience of them but others seem to rate them highly.
Korail have what at first sight looks like a good website, but there are several shortcomings worth knowing about. You can check schedules and seat availability but to make a reservation you have to become a member. Nowhere on the site does it explain that you need to go to your local train station with your Alien registration card and pay them 8,000 Won for a year's membership.
Armed with your membership number you can make reservations on the website, but again, unless you have a Korean issued credit card you cannot pay for it online. At least with this site you can book your seats online and then pay at the local station giving them the reference number of the booking.
The website cannot handle return journeys, so you have to re-enter the details to check the return leg which is tedious. More significantly it requires that you know which station in Seoul the trains to your destination leave from. Try searching for a train from Seoul to Yeosu and you get no results, that's because the trains leave from Yongsan, a couple of miles down the line from Seoul Station. Similarly if you'd like a train to a destination on the line to Chuncheon you have to know that it starts from Cheongnyangni.
We have rented a car once through Hertz with no problems. The most important stipulation is that you have an International Driving Licence.
There are six intercity bus terminals in Seoul. The smallest is Sinchon Bus terminal, which is very easy to miss, especially if there are no buses there, as it is simply a converted container with a couple of ticket machines inside. The other five: Central City Terminal, Dong Seoul Bus Terminal, Seoul Nambu Terminal, Sangbong Bus Terminal are much larger and easier to find. They offer varying levels of services to all cities and large towns in the country.
On the few trips I have made I've been unable to pay by credit card and unable to buy return tickets, so be prepared with cash and don't be surprised if you have to buy your return ticket at the other end.
The Tour2Korea website has a useful feature to find out which bus terminal you need for your chosen destination, but this time the limiting factor is that you need to know which province it's in.