We can't tell you the cheapest tickets, because no-one has devised a system that will calculate it (though people are working on it) however you can have a go at finding out the cheapest combination of tickets by looking at purchasing a combination of tickets "splitting" at various stations en-route.
Obviously it takes a long time to look up every combination, however being familar with the area, I can suggest buying a combination of Scarborough - York; York - Northallerton; Northallerton - Darlington and Darlington - Durham tickets, all routed Any Permitted except the last one which is XC Only. That does save a few quid. I will not say this is the "cheapest" set of tickets though, something involving GC and Eaglescliffe may be cheaper. The problem of course, with attempting to find the cheapest tickets is that it is questionable if the saving is really worth the extra journey time sometimes - but that's subjective!
Note that the train will need to call at the station(s) where you change from one ticket to the next.
The cheapest ticket is a Super Off Peak Return, at £34.30 but this price can be reduced by splitting the journey into a combination of tickets as suggested above.
Another way to save money is by taking advantage of routeing anomalies and starting/finishing short, however such anomalies can be withdrawn shortly after they are revealed. I can't really think of any that would apply here anyway.
