Try the Long Bar, it's two mins walk from the central station. Good place to start, you can even win free rounds on a roulette game thing. Also the Witherspoon place is just up the road after that the quayside is a stroll.
Newcastle's not bad at all, even though they changed the feckin Football ground's name from St James Park. Anyway after landing from Dubai I holed up there for a few days and promised myself I'd go back, despite the Pub near the Tyne Bridge knowing fark all about their own area. Generally good memories of the people and shit loads to do at night and weekends. Lovely.
The Bigg Market WAS the place to be for 18-25 yr olds in the early 90s, then the scene moved down to the Quayside which is pretty decent but of late the Gate has opened and the area around Central Station is really popular. No ones mentioned the Haymarket area either - thats fairly lively.
All of these places are less than a 10min stroll from each other, or a cheap taxi if you don't know the area.