Mercado San Camilo (on San Camilo between Nicolás de Piérola and Perú) is the main market in the city, where you can find meat, vegetables, fruits, good cheese, and anything you can think of. There are also juguerías, or juice stands, where you can drink any kind of mix for US$1.50.
On the Plaza de Armas [1], El Super (Portal de la Municipalidad 130, tel. 054/28-6616, 9:15 a.m.–9:15 p.m. Mon.–Sat., 9:15 a.m.–1:30 p.m. Sun.) can fill in with dry goods, yogurt, tuna, beer, and wine.
Links:
[1] http://www.moon.com/destinations/peru/lake-titicaca-and-canyon-country/arequipa/sights/plaza-de-armas