From Khon Kaen to Nong Khai, we takes about 1.30 hours by car.
By bus
There are departures to Udon Thaini at least once per hour from the BKS station on the main drag on Prajak road. The hour-long ride costs 20 baht in 3rd class (non-air con).
By train
You can go to Nong Khai by train. It would take about 10-12 hours. A first class sleeper ticket from Bangkok to Nong Khai or vice versa is currently about 1200 baht, and a second class sleeper ticket (not bad for the price) is 748 baht ($21.43 USD as of Jan 2009). Sleepers often sell out at peak times so you may need to book in advance.