As more accommodation choices offer better dining options to both residents and non-residents alike, the once popular idea of self catering seems to be replaced by half-board, all-inclusive or at least dining out each night. For a couple there's a vast choice of economic eateries around the resorts meaning self catering doesn't always save you that much money but for a family eating in it can make a difference when budgeting your holiday.
The most common option for dining in tends to be at breakfast time and oddly enough many tourists bring food from home (especially us Brits with our bacon, sausages, tea bags and coffee). No need, you can get it all here. For typical foods from back home check the supermarkets in-resort. They are a little bit more expensive than their counterparts in residential areas but will stock many international food items.
If you want to do a big shop for the week then head out of the main resort, find where the locals shop. In Puerto Rico / Amadores that will be to the Spar or Hiperdino supermarkets in Arguineguin (5 minutes in a taxi) and for Maspalomas and Playa del Ingles that will be either San Fernando (Spar, Hiperdino, Mercadona or Lidl) or the big shopping center at El Tablero for the Hiperdino there. Buy all your fags and booze there too, they'll probably be cheaper than back in the airport.
You can find stores and check times (even prices) with the help of google translate and the following web sites :-