There is a railway line that runs from Udon Thani all the way down to Bangkok and for the first part up until Kumphawapi, it runs almost alongside Highway 2 (road to Khon Kaen). However, it doesn't go that close to the main area of Kumphawapi, I know of one station but it's just one building and a platform. There's no public transport to talk of up in that region, you will have to hire a car if you want to get around with relative ease. Once out of Udon itself, the roads are easy to navigate and are very quiet most of the time.
They drive on the left, same as the UK but not the same for the rest of Western Europe. If you are up to trying left hand drive then your best option is to hire. You can get buses from Udon down to Kumphawapi, but they are not the type of buses we would associate with long journeys, they are Songthaews - basically a pickup truck with two bench seats in the back facing each other with room for 5/6 people on each side, plus of course standing room. Most of the time these are full up with locals as they don't run all that often. If you are with a local (Thai) and that person knows their way around plus you are the adventurous type, then try the buses you might find it a great way to see Issan. I have no idea where you would get information about routes, times etc and wouldn't even know where to suggest looking. There probably isn't a website, well not in English anyway, that would give out this information.
If you want a hassle free trip then a car is a must, they have car hire places at the airport, but it's best to book in advance.
Have fun
