We will be in York in mid October this year for three nights, then moving east with the next night in Pickering. Originally, I had planned to stop in Castle Howard on the way to Pickering, then catching the Noon steam train to Whitby, spend an afternoon in Whitby, return to Pickering for the evening (we are booked in a B&B there for one night).
After some study of what I find on the internet about Castle Howard, I suspect that this plan would not give us enough time at Castle Howard.
Another option is to spend 3 or 4 hours at Castle Howard (is that suitable to see the premises?) then drive on to Pickering, spend the night there, get up early the next morning and take the 9:25AM train to Whitby(arriving 11AM), return to Pickering on the 2PM train(14:00) - arriving at 3:30PM (15:40). We could take then 3PM train and arrive at 4:55PM. Our next city to visit is Durham, which is at least two hours or more of driving.
A third option is similar to the second, is after spending the night in Pickering, drive to Whitby instead of experiencing the train, then drive from Whitby to Durham.
A side questions becomes, just how nice is the train ride vs. our driving, and our missing driving through the Yorkshire Dales to Durham?