I see many of my favourites on here (although I prefer Sugo in Altrincham as it has a friendlier, more intimate feel so probably wouldn’t prioritise the ancoats restaurant over some of the others mentioned here myself). I would add:
Pasta Factory (freshly made pasta) and its sister pizzeria Noi Quattro. Both of these are great in general, but especially for vegetarians and vegans. Their sauces are more inventive and tasty than just adding pretend cheese, although they have started making their own vegan cheeses.
The latter is very close to the Arts and Crafts Centre, which is a must-visit if you like jewellery, ceramics, etc.
Best cocktails are at El Gato and Refuge, which are also great for a more upmarket meal. Refuge does a lovely Sunday lunch.
Cheap eats - I’d add the Arndale Market (not very pretty, but some very nice food), Siam Smiles and the strip of Vietnamese cafes on Oldham Road, Ancoats.
Cultural stuff - guided tours by Jonathan Schofield are a really good bet. If you google him, you’ll see what’s on when you’re in town.