Not sure what the GNE fares are but here's some fares from Arriva...
Middlesbrough-Newcastle £5.70 (rtn £6.70)
Middlesbrough-Durham £5.20 (rtn £6.70)
Durham-Newcastle £3.50 (rtn £5.80)
Durham-Chester £2.70 (rtn £4.80)
Chester-Newcastle £2.70 (rtn £4.80)
RE: Arriva North East