There are quite a number of running/athletic clubs in the Newcastle area. Some are purely running based, whilst others (generally the larger ones) include field athletics - all the jumpers and throwers etc. Here are a few for starters. I can get more if you need them.
Claremont Road Runners ----
www.claremontrr.freeserve.co.uk
Low Fell Running Club ----
www.lowfellrunningclub.co.uk
Elswick Harriers ----
www.newnet.org.uk/elswickharriers
Gateshead Harriers ----
www.gateshead-harriers.net
I'm expecting other members to maybe add to your choice before too long.
Good luck, Rodger.