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.