You can drive to Tibooburra from Bourke, although there is little to see other than the wonderful desert landscape and wildlife. Near Tibooburra is the old gold town Milparinka consists of several old buildings and nothing else. Behind the two places is the Sturt National Park stretching upto the South Australia and Queensland borders.
You don't have to have a 4WD to access the majority of the park, although be sensible and inform the ranger of your actions. The target for most modern day explorers is to reach Cameron Corner where all three states (New South Wales, Queensland and South Australia) meet. It is a long way out but an amazing experience to be there.