If you don’t have a car here are some of the transport options to Dubbo.
The Country link XPT line goes through Sydney, Katoomba, Lithgow, Bathurst, Orange and when it gets to Dubbo it turns around and heads back. It arrives in Dubbo every day about 1:45pm and leaves at 3 PM. For more information check out this website: http://www.nswtrainlink.info/timetables
As a broad generalisation, to get to Dubbo from most other places in NSW by train requires you to travel into Sydney then out to Dubbo from there.
Bus services are limited, but you can get from Dubbo from further out west (Warren, Nyngan, Narromine, Lightning Ridge, Gilgandra, Bourke etc) by bus. http://www.nswtrainlink.info/timetables
Sid Fogg runs a coach from Dubbo to Newcastle via Mudgee.
Qantas and Rex run a regular service between Dubbo and Sydney on weekdays ($150-$200 if you book early enough) but it’s a bit sparse on weekends. Qantas also have flights to Brisbane.
Pelican (flypelican.com.au) have flights from Dubbo to Newcastle and from 2017 to Canberra.
For a list of all upcoming flights in and out of Dubbo check out this site.