Changing once at Manchester, using Transpenine Express for both trains 1 adult costs £45.40/1 child costs £22.70 for a one way anytime ticket for travel on any services on that route on the day of travel. The journey (including changing trains) takes around 3hrs 30mins. Return tickets cost about £8 more.
There are discounts if your ticket is purchased in advance and it's for a specific train and also if there are railcards involved. For more information on that check National Rail's website which I have attached a link for below.