From Robin Hood's Bay, we took a bus to Scarborough. From Scarborough, we took a train to York. From York, we took a train to King's Cross Station in London.
In London, we took the Piccadilly Line (the Tube Map app came in handy) to Terminal 4 of the airport, where our hotel is. It's a lot of trains, but it is all rather easy - except for getting a forward facing seat on trains :(