I did the same journey last year for an interview/open day..
Once you fly into Newcastle and come into the main 'lobby' or whatever you call it, go left through some nice automatic doors. This takes you to the metro station. Don't walk outside and walk around aimlessly like me! Don't be fooled by the lack of barriers ala london underground, they do actually check tickets! I found out the hard way..
The metro will take you to newcastle central station, then from there you will need to find the train to durham. Sorry I can't remember train/metro prices, but it wasn't bank breaking!
I got a taxi from Durham station to Aidans, to be honest if you don't know the way its probally best. Cost me £4.50 I think, again hardly bank breaking! To get the taxis probally have to go under the underpass to the opposite side of the tracks, iirc?