Wynyard Woodland Park
Formerly known as the Castle Eden Walkway Country Park this park in Billingham has numerous woodland footpaths and a visitor centre. National Cycle Network Route 1 also passes through the park so you can bring your bike too. Just north of the park is the Thorpe Wood Local Nature Reserve which offers a chance to extend your outing. There's also a delightful wildflower meadow, a fine cafe and a fascinating planetarium and observatory.
The Castle Eden Walkway rail trail runs through the park so you can continue along this to Thorpe Wood and Brierley Wood.
Also nearby is Cowpen Bewley Woodland Park where there's pretty lakes and lots of birdlife to see. Near Stockton you can visit Coatham Woods with two waymarked woodland trails to try.