There are no trains to Richmond any longer but there is a regular 15 minute frequency bus service (30 minutes on Sundays) from Darlington - the buses stop outside the railway station and the connection is therefore very good. The journey takes just 30 minutes by these express buses - once in Richmond they stop in the main square. The service numbers are X26 and X27 - see http://getdown.org.uk/bus/bus/x26.htm for the timetable.
From Richmond there are regular buses in to the Dales - you can easily get to Reeth, Leyburn, Masham , Hawes, Aysgartgh and many other places by local bus - you just need to do some careful planning using the link below. Or ask for more help in the travel forum here on VT.
It has to be said that public transport around the Yorkshire Dales leaves something to wish and you are much better off with a car, both for getting here and going around. If you don't have a car, take the train to Northallerton (London to Edinburgh line) and change for irregular buses. Alternatively, try a way of getting to Wensleydale and take a taxi from for instance Leyburn.