If you like pubs, after a hard day's travel, get yourself into the Bridge Hotel. There are a few window seats in there that overlook the magnificent High Level Bridge, that you can walk over at the road level. Or, get a train to Wylam along the Tyne Valley (Sit on the right hand side leaving Newcastle) and get a visit to George Stephensons birthplace house. It's not too far from there to have a wander across the beautiful Hagg Bank railway bridge, a pioneering 240ft single span rib arch built 1876. The Boathouse pub almost adjoins Wylam station, has fabulous pies and excellent real ale.
Or perhaps jump on an X30 bus and pay a visit to Causey Arch, supposedly the worlds oldest railway bridge, alongside the nearby Tanfield Railway?
Definitely visit the 'Centurion' bar in Central station, even if just for a soft drink, the tilework inside is stunning. Then there is the Stephenson Railway museum to visit, or perhaps the discovery museum where they have the 1st ever steam turbine powered ship Turbinia is your interests extend beyond railways. Also reachable via a bus ride on service 56 is the NELSAM air sea and land museum, a fascinating place with a Vulcan, a Blackppol tram and various army vehicles amongst others! On the 56 you will also pass the Bowes Railway Museum near Springwell, booked tours only but one of few remaining rope worked railways existing. Or just buy a Tyne Tees day ranger and visit Shildon and Darlington museums then take a ride along the bits of the original Stockton and Darlington railway on a lovely 158 or if you are lucky a 156

Hope this is of use!