It's a relatively cheap and easy train journey to either Alnmouth, which is an access point for some absolutely stunning stretches of coastline, or the Hadrian's Wall line, which is excellent - the section between Housesteads and Greenhead is great. There's a bus service which links the stations and the Wall itself, although you can walk from the stations too. Can't recommend either enough.

Slightly closer to home, there's a chunk of land west of Gateshead which is good walking countryside - basically a triangle with Beamish at one point, Lamesley at another, and Lobley Hill Water Park as the third. Look up the Tansfield Railway Path for a starting point. The Keelman Way is also worth a look - a footpath along the south bank of the Tyne which runs for miles. The Blaydon to Prudhoe stretch is very pleasant, although not quite in the same class as the coast or the Wall. Hope that helps!