The Great Cobar Museum is the right first stop in town. It occupies the former administration building of the Great Cobar Copper Mine, dating from around 1910, and uses that setting to tell the mining, pastoral, and everyday history of Cobar.
The collection is broad rather than polished, which suits the place. Look for the material on early mining, the practical life of isolated outback families, and the way Cobar kept reinventing itself around metal, water, and distance.