Is there a direct bus from Korat to Sukhotai or is it better to go back to Bangkok and take a bus from there ?

Using NCA just ask at your Bus terminal for direct route. Rrom Korat I would go to Ayutthaya and then to Pitsanulok and then SukhotaiIts a long way...whichever way you go. Have a good trip

There is a direct bus from Korat to Pitsanulok, because I've taken it. It probably is the same one which goes on to Chiangmai. From Pi-lok to Sukhothai is only an hour or so.