Depending on your budget and what your after you could try The Snowy Owl Pub & Restaurant in Cramlington - Vintage Inns about 15 mins drive from Gosforth, we go here for a pub meet once a year an the food is very good. There is some country lanes to the west where a lot of bikers go on a weekend. At the other end of the scale is http://www.matfenhall.com/ which is about a 30 min drive to Gosforth but well out in the sticks with some good biking roads.
+1 for The Snowy Owl, properly nice food in a decent setting with good staff too
Hamsterly Forest has some good downhill and a bit of “Northshore” style trails (allbeit on a tiny scale)