There are many options where you can buy food and whatever you need. Since a small shop until supermarkets, you can choose where is more convenient for you.
Some of the more importants supermarkets in Mexico City, are the following.
Furthermore, in Mexico City exists a lot of local markets, where you can buy fresh and good food by a lower price. You just have to look for a market in a GPS. Near to the CIC there are two options:
Mi Mercado Lindavista
Mi Mercado San Bartolo
Also, depending on the neighborhood in which you live, one day a week there are usually small markets on wheels that are an excellent option to buy what you need. These are called "Tianguis" and you can ask to your neighbors when can you find them.