Patio del Ekeko
There are some interesting shops geared towards tourists, including Patio del Ekeko, free WiFi included. The old Jesuit convent, half a block from the Plaza de Armas, is free to enter and has very good shops for fine Alpaca garments and local products. The place itself is very nice.
La CompañÃa
There is also a square near La CompañÃa having a lot of the more expensive handicrafts shops.
Arequipa is called the World´s capital of the Alpaca, a wool that comes from this Andean camels and considered very fine and valuable. Big factories and several boutiques offer fine products at great prices. A good quality alpaca sweater will be between 70 to 180 soles according to colors and designs. Be aware some handicraft stores offer very cheap alpaca, unfortunately fake ones.
Most typical handicrafts come from the Chivay area. Colorful embroidery called Maquinaza style, representing daily activities or nature scenes.