Nakorn Ratchasima is something of a transportation hub for Northeastern Thailand, which means there are quite a few options for getting there. Unfortunately, flying isn't one of the options. Commercial flights have been tried in the past, but apparently they weren't viable, since Korat is just too close to Bangkok.
This leaves you with the options of bus or train, unless you want to opt for a private transfer. Buses leave Bangkok's New Mor Chit station for Nakorn Ratchasima practically every few minutes. The trip takes around four hours.
Train: The city is where the two Northeastern rail lines split, so all trains to Nong Khai and Ubon Ratchathani pass through Nakorn Ratchasima.