Dubbo is accessible from Newcastle (via the New England and Golden Highways). Approcimately four hour drive.
Countrylink operates several train and bus connections between Sydney and Dubbo every day: there is a direct train service once a day, and a number of bus connections meeting trains from Sydney terminating at Lithgow.
Check out http://dubbo.org for a list of things to do in Dubbo.
Here's a sample:
Farmer’s Markets There are Farmer’s Markets every month in Dubbo, NSW. They are on the first and third Saturday of every Month, near the Tourist Information Centre. Fresh Produce, poultry, coffee, jams, preserves, Olive Oil, sourdough bread, etc.
Ampalee Bed and Breakfast (http://ampallee.com.au/index.html) - Self contained units located outside of Dubbo Proper. $150+$20 each over 2 people. Dubbo Cabins and Caravan Park (http://www.dubboparkland.com.au/) -
DubboOnline.com - Online guide to the City of Dubbo