Travel to Whitby by rail from London.Or is it better to go to Scarborough and get a bus?

Yeah, Whitby is quite a journey by train, especially coming from the south. By train, Scarborough would be the better choice.

Alternatively, you could get the bus from York. It's one of the best bus journeys in the country - not quick but very scenic.

https://bustimes.org/services/840-leeds-whitby

If it were me, I'd take the train from London to York, stay a couple days there, then bus to Whitby. However, I just love York and use any excuse to stay there a couple days.

Thanks for the information about the bus from York, gives another option.

I love York too. But, I've not been to Whitby before, so I was planning on a couple of nights stay there, before my walking trip.

I spent 5 days in Whitby this September. From London, I spent 2 nights in York (it’s awesome). I purposely did not book my travel to Whitby from York until the night before I left. I ended up taking the train to Scarborough and took the bus to Whitby from there. On my return, I took the train all the way back to London. Of the two, I preferred the train journey. I am hoping to visit Whitby again and would likely combine that with another stopover in York. However, I would look to take the train from York to Whitby, likely via the steam train, or a bus from York. Going to Scarborough by train with a connection by bus to Whitby was fine and easy enough (the bus stop to Whitby is right outside the train station). On an earlier vacation, I day tripped to Scarborough and felt it was just okay, although getting out of town to walk The Cleveland Way was the saving grace for that day trip. When I was there in September, the busses were running behind as there were wildfires in the area that were winding down. So, I spent more time than I was planning to at the bus stop, waiting for the bus. Still, I would do the train to Scarborough with a bus to Whitby again, but it would be my third option. You will love Whitby.

There are more ways than the obvious to do this. I'm used to arriving from the north due to where I live, but those northern routes can be adapted.

The train service to Whitby is thin- but from KC is train to York, change for hourly TPE to Middlesbrough, then train to Whitby. Because the train service to Whitby is thin it maybe works more smoothly on the way home.
Sometimes you can reduce that to two trains (one connection) by using a Grand Central Sunderland bound service to Thornaby or Stockton and changing into the Whitby train there, as some of them start back at Newcastle.
However you could take the 10 minute or so walk from Middlesbrough railway station to the bus station then the hourly X93 Scarborough bound bus over the Moors. Another very scenic run.
Or the every 30 minutes X4 bus via the Cleveland Coast, again very scenic.

The arguably better way to join the X4 is LNER to Darlington, then the hourly Northern train to Saltburn. The X4 stops right outside Saltburn railway station. Saltburn is a seaside resort in it's own right, and could be a base for a couple of nights to explore the Cleveland Coast.

Another possible stop over is Grand Central to Hartlepool, for the National Museum of the Royal Navy and the memorials to the Bombardment of Hartlepool (with significant loss of Civilian life) in WW1, then train to Whitby via Middlesbrough. Hartlepool could also be used as a base for a bit of walking on the much neglected Durham Coast- now restored to nature after it's industrial and coal mining past.