Barnard Castle is situated close to the A66 trans-pennine road on the A67. The A688 provides a link into the town from northeast County Durham.
Be warned that there is a narrow bridge on the approach from the southwest on the A67 that is not suitable for heavy vehicles. The bridge also involves a 90 degree turn on both entry and exit and it is not unknown for unwary drivers to catch the bridge's side walls. Traffic lights control the flow due to the single-lane nature of the bridge.
There is a large "pay and display" car park at the Morrisons store. There is also free parking on the cobbles in the town center, however Wednesdays and Saturday mornings are market days and you can't park here. Generally though, you should have no problem finding a parking space in the town, and exploring on foot.
Public transport is limited to local buses within in County Durham and the Blackpool to Newcastle coach service provided by Classic Coaches.
Barnard Castle is small enough that the town is best explored on foot.