miles,

bilbo is wrong about the trains.

If you need to change trains it is often CHEAPER to buy two single tickets than one ticket that covers both trips.
It often works out cheaper to buy two singles than a return

Use this
National Rail Enquiries - Fare Finder

harrogate to london £42.50 single
harrogate to york £6.60
york to london £22.00

They take the piss 42.50 to 28.60!! You will be restricted on the times you can travel though, if you're not too fussed you'll be making a big saving.