Ok, assuming you get hold of a street map (should be one available from tourist info at the station) to find out where all these streets are:
Well, THE thing to do, of course is to go to Bettys, on St Helens Square. But there's generally a queue, especially at this time of year, Christmas shoppers and so on. There is a less well known branch, Little Bettys, in Stonegate round the corner, there is often a queue there as well, but maybe less so.
Otherwise, a teashop I've used with visitors (and which I think is open on Sundays, but am not certain) is Bullivants on Blake Street (it's opposite MacDonalds.) Otherwise, there are loads of others, and all the usual coffee chains etc.
Pubs... Well, there are so many. Anything by the York Brewery is good - The Three Legged Mare in High Petergate, The Last Drop in Kings Square (I think that's what it's called) or my favourite of the three, The Yorkshire Terrier in Stonegate (Does food at weekends I think) There is a good table in the window upstairs, if you like people watching.... For the best ranges of real beer, the Maltings, just off Lendal Bridge on the other side of the river from the Minster (also does food at lunchtimes, be warned the chilli is very hot!) or The Blue Bell in Fossgate (does food some lunchtimes I think, but not sure), which is tiny and very original. For good food, and a real 'local' feel, the Royal Oak in Goodramgate. It's my local.
(These are all to my taste of course, which is 'real ale, food sometimes, not too much in the way of loud music or fruitmachines or yoofs')
Many tourists go to the Kings Arms on the riverside, and I haven't been down to the river lately, but if we have much more rain, that one will be under water... If the river is up, the Ouse Bridge is a good place to stand and see the full effect...