Depends - are you only asking for the best routes, that is, the fastest/least busy/most comfortable journey, or are you also looking to save money on buying a ticket? For example, a Durham to Manchester Super-Off Peak Return is valid via Blackburn, with a break of journey - so you can use it to travel to Blackburn and then to travel from Blackburn to Manchester - and is £3 cheaper than a Durham to Blackburn ticket.
If it is just routes then the journey is not very complicated: travel from Durham to York or Leeds - all southbound trains stop at York, whilst Transpennine Express trains or Cross Country trains towards the South West (NOT the trains which terminate at Reading/Southampton) go to Leeds. Then, take a Northern Rail train in the direction of Blackpool North, which will take you to Blackburn. Blackburn to Manchester is a direct train, takes 50 minutes. Manchester to Durham can be done directly, at 21 minutes past the hour every hour, or just take any TransPennine Express service that goes via York (those to Hull, Scarborough, Middlesborough) and change for Durham there.