We visit Manchester fairly regularly since our daughter has settled there, and our favourite tea/coffee place is Pot Kettle Black in the Barton Arcade on Deansgate. It is right behind a Spanish bar/restaurant that faces onto Deansgate called Lunya. We liked the food in Lunya, but the service wasn't terrific.
If you're into fancy teas, my wife and daughter both love a tea shop opposite the cathedral, called Proper Tea, but I find it a bit pretentious.
I may be boring, but the most enjoyable meals I've had in Manchester have been at Jamie's Italian, which is in a magnificent old bank building that is worth a visit just to use the fabulous loos.