Cafe 21 is good, owned by Terry Laybourne, he also owns Jesmond dene house, which is the best restaurant in town, but mega bucks to eat there.
Best Italian IMO is Secco, its pretty much the only Italian in Newcastle that really cooks proper, interesting, regional Italian food Secco linky
Just outside newcastle centre , in Sandyford, there’s a hidden Gem called Caffe Zonzo, again italian, small menu, really great atmosphere, and great staff.
As mentioned above Pan Haggertys gets good reviews, and I’ve only heard good things about it, but never been.
Blackfriars is an incredible building, but food is a bit hit and miss in my experience.
there’s a good tapas bar near St James Park called El Coto, and as others have said, Barn Asia is very good
There’s not much choice in Newcastle for really good food, never really understood why TBH, no good steak houses, or French….
EDIT: don’t go to Flat Bread Cafe, as mentioned by CaptJon, its closed down