The daily XPT service from Sydney to Dubbo has a connecting coach service to Narromine. Book the entire route in advance.
Narromine is a well developed airport for a town of it's size. It's possible to charter or fly yourself there, but there is no regular passenger transport at the airport. Dubbo is the nearest airport, and there is a weekday bus connection from Dubbo to Narromine.
Gliding. Narromine is one of the premier gliding centres in NSW