From Durham Bus Station, catch the 6 or 6a to Bishop Auckland and get off at the stop for the Botanic Garden. Cross the road at the pedestrian crossing and walk along Elvet Hill Road, taking your first left on to Mill Hill Lane. You will see the Museum on your right, just follow the path around to the entrance. This is a five minute walk of 0.2 miles.
From Durham Railway Station, you can walk down the hill from the station to North Road and take a bus to the Museum as per the directions above.
Walking to the Museum from the bus/train station
From the train station, make your way down the hill to North Road where the bus station is, then follow North Road down towards Framwellgate Bridge. Before reaching the bridge go uphill, taking the first left past The Fighting Cocks public house to continue up cobbled South Street to reach Quarryheads Lane. Here, turn left and walk for a few minutes to reach a roundabout. Turn right here, going up Potters Bank until reaching Elvet Hill Road on the left. Walk along Elvet Hill Road until you reach the Oriental Museum on the right. Please note that this is a hilly route, which takes approximately 25-30 minutes.
Alternatively, at Framwellgate Bridge, just before crossing the bridge, take the steps down to the riverbank path and follow the scenic riverside path up to Prebend's Bridge. Here, turn right up the hill and then left at the stone cottage. Continue through the white gate to reach the roundabout and then continue as above.