What would be the best way to get from LHR to Whitley Bridge by train? First time in the UK.
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Whitley Bridge is a rural station in Yorkshire, near Leeds, about 250 km north of London. According to Wikipedia it is only served by a few trains a day, so unless you are traveling at exactly the right time, you'll need to take a taxi or local bus from a nearby station with better service, such as Selby, Knottingley or one of the Pontefract stations.
If you want to go by train all the way, you're looking at a travel time of at least 4-5 hours, and it won't be particularly cheap. It might be quicker and cheaper to extend your air ticket with a domestic flight to Leeds-Bradford Airport and take local transportation from there.
By train, you would need to take a train from London Kings Cross and change to a local train somewhere around Leeds or Wakefield. See this question for getting from Heathrow to Kings Cross, and use the National Rail journey planner to find a suitable connection from there.
(If you enter "Heathrow" as a departure point in the National Rail planner, it will try to put you on the expensive Heathrow Express into central-ish London. It is cheaper and not much slower to take the tube instead).
Note that the train ticket can be significantly cheaper if you buy it in advance, but then it will be tied to a particular departure from Kings Cross, and you won't have much flexibility to deal with flight delays.