Good pubs mentioned above - all will be more expensive than 'spoons, but not too bad - though there's Wetherspoons/Lloyds on the Quayside - The Quayside - and near Central Station - The Union Rooms and The Milecastle; The Centurion (at the station) is also fairly cheap.
Prices vary wildly at most pubs/bars depending what you drink - stick to bog-standard lagers, Wylam brewery ales, and house spirits and avoid craft beers/premium imported lagers and spirits if you want a cheap night. If you do decide to drink something fancier then the Bridge Tavern has the best craft beers (better than BrewDog), Tilley's has a massive choice of beers, Cumberland Arms is good for cider, Alvinos for cocktails. Anywhere on Dean St or Collingwood St (the "Diamond Strip") is stupidly pricey whatever you're drinking (and aimed either at hipsters or Geordie Shore types so best avoided).
Bigg Market area is generally very cheap, lots of offers, level of surrounding carnage depends on what bars you go in, what day of the week it is, and what time of night (its also a surprisingly good place for a sit-down meal - there's Italian/Vietnamese/Greek/Indian/Lebanese restaurants and pub grub, all fairly good, reasonably priced, and serving food on proper plates not recycled bits of wood).
Head of Steam is friendly, conveniently placed opposite the station for public transport/taxis, plays good music downstairs at weekends, and its open late, so its a good place to end up before working out if its home or clubbing or more of the same.