Hmm, I would like to recommend you places in York but much of what I know is on the touristy side. D: Although I really feel that everyone should go to the Roman Baths Pub on principle of it is a pub right on top of the remains of the Roman garrison's baths, which is touristy but in a low-key way.

Also there is a small restaurant on Gillygate called Asia Gourmet which has a prawn tempura with cheese roll that is EXCELLENT, 100% recommend.