Great North Museum is good. The Discovery Museum has Stephenson's Rocket until September (both are free but you need a ticket for the Discovery).
Castle Keep, obviously. Take a walk through Exhibition Park and have a pint at Wylam Brewery and get down to the Ouseburn where there are some mint pubs (Cluny, Tyne Bar, Cumberland, etc). Jesmond Dene lush for a walk too.

