There are many language schools in Arequipa [1]. Some offer group lessons and other individual classes. If you are interested in a homestay, ask the school if it arranges them. Many do. A handful of recommended schools are listed here. Contact Carmen Cornejo at the Centro de Intercambio Cultural (Cercado, Urb. Universiteria, tel. 054/22-1165, www.ceicaperu.com [2]).
Besides Spanish, Centro de Idiomas UNSA (San Agustín 106, tel. 054/24-7524,) offers English, French, Portuguese, and Chinese classes. Rocio language classes (Melgar 209, of. 22, tel. 054/22-4568, www.spanish-peru.com [3]) is a more affordable option, and classes are catered to students’ flexibility.
The best is Centro Cultural Peruano Norteamericano (Melgar 109, tel. 054/80-1022), which offers Spanish and English courses and has offices throughout Peru.
Links:
[1] http://www.moon.com/destinations/peru/lake-titicaca-and-canyon-country/arequipa
[2] http://www.ceicaperu.com
[3] http://www.spanish-peru.com