Quick answer: You buy food at the grocery store or market. A convenience store sells quick beverages and snacks (to go). Mc Donald's is a good example of a fast food restaurant. A restaurant is a sit down place where you take your time eating and enjoying your meal. April's answers are correct about Mc Donald's and donuts being sold at donut shops.

Does it solely serve doughnuts/chicken wings/that kind of food? If so, then it could be classed as a 'fast food restaurant' (like McDonalds), the proper name is 'quick service restaurant' but that's not particularly colloquial. Alternatively, if a store only sold doughnuts (like Dunkin' Donuts) then it could be called a 'doughnut shop' or 'doughnut store'.

It would help to see a photograph of what you have in mind! Does it have seating? Is it a self-contained building with an entrance that opens onto the street or a walkway, or is it a small business with only a serving window? A small business with a serving window and a few stools, such as you find in many shopping malls, I would call a snack bar. I would only call it a restaurant if it had tables and chairs - usually inside, but sometimes just on the street - and served proper meals.