Main recommendation would be an afternoon or evening drinking around Ouseburn, especially if the weather’s good. The Free Trade Inn is the best pub in the world, Cumberland Arms not far behind. The Cluny, Tyne Bar, Arch2 (new Microbrewery) also worth visiting.
Can visit the urban farm and the Biscuit Factory (arts space thing) if you’re that way inclined.
If you want a night out, worth checking what’s on at World Headquarters or Ernest.
Good food options around there, too. Tower Café Pizza cheap and decent. The Bake (Lebanese) really good. Both of those BYOB. I’ve not been to Cook House, but it’s a small reservation-only place in a shipping container people rave about. Kiln also sounds good.
Otherwise -
-Chilli Padi in city centre, or Cal’s Own (Jesmond) for food.
-Laing art gallery, Baltic, Side Gallery.
-General quayside walk (past millenium bridge down to free trade)
-Quayside market on sunday morning
-The Bridge Tavern and Crown Posada down that Tyne Bridge end of Quayside
-Metro out to the coast if weather’s nice. Tynemouth especially
-Staiths and Staiths café on the Gateshead side
lots more beer and pub recommendations but that’s probably enough if youre only here a couple of days!