Travel from York to Conisborough

Taking a bus will literally take you for ever- probably 5 or 6 hours each way.

Take a train from York to Doncaster (20 minute journey), and change there for a Sheffield bound train - 1 stop, 10 minute train journey. 1/2 mile walk to the Castle.
Or go across the road from the Railway station (covered walk) to the Interchange and catch a frequent bus routes 220 to 222 and X78. But through train journey seems simplest.
Depending on connections can be as short as a 40 minute overall journey, or as long as 1 1/4 hours each way.
Trains run until very late, so you won't get stuck. Buy your ticket on the day at York Station- £23.70 off peak day return

The Castle is open 10 to 5 daily, £8.50 admission.

You could take a bus - but it would need two changes and take 3 hours.

Much easier is a train to Doncaster then 221 or X20 bus. Both would put you a couple of minutes walk out. Less than an hour.

You could train the whole way - change at Doncaster for Conisborough, but it's a longer walk and overall saves just a few minutes

I have never heard of a 2 change route taking 3 hours. Traveline (which is the usual bus planning tool) gives 4 1/2 to 5 hours journey time via various route permutations, but usually involving Leeds, Wakefield and Barnsley among other places.
The only way to get the bus journey down to around 3 to 3 1/2 hours is to use National Express or Megabus for part of the journey which needs pre-booking.

Thank you for your suggestions. I like easier travel, so will probably try the train to Doncaster. I assume the return trip to York would be the reverse?

Yes, same in reserve. Use the LNER website to plan your journey more exactly.