Harrogate is easy to get to by train from either Leeds or York (both main-line stations). You rarely have to wait longer than 30 minutes to an hour between trains (depending on the time of day of course!).
Harrogate has trains to Leeds and York. Buses go to Leeds , Ripon, Borobridge, Patley bridge and York. The service to York is new for 2008, the first direct bus for years. [X54 ] takes one hour.