X26/X27 Bus to Richmond. From Richmond you can connect to the Dales & District 159 Bus to Ripon, which stops in Leyburn. From Leyburn you can connect to the Little White Bus Wensleydale Voyager 156 Bus, which stops in Redmire and in Castle Bolton right by the castle. This last is not that frequent and does not run on Sundays.
Alternatively, there is a single service which connects direct from Darlington to Leyburn. The Procters 610R Bus is the Darlington College service into the Dales, so again runs only Monday-Friday, but there is a single service a day each way – 5:25 PM from Darlington to Leyburn, and 7:44 AM from Leyburn to Darlington. The journey takes about one hour and leaves from Darlington Town Hall.
Dales & District 73 Bus from County Hall in Northallerton (just over the road from the train station) to Bedale. From Bedale you can connect to the Little White Bus 155 Bus to Leyburn, where you can again connect to the 156 to Redmire and Castle Bolton.
On Sundays only, you can take the Little White Bus 856 Wensleydale Flyer to travel direct between Leyburn and Northallerton.
On Saturday and Sunday this year, there is also the option of taking the Wensleydale Railway. You can use this to travel between Redmire (for the Castle) on the one hand, and Leyburn or Bedale on the other, and thus connect to the 856 Wensleydale Flyer.