If all you truly need are a bed, a stove, etc., you'll be better off renting unfurnished, supplying your own, and then selling them when and if you move on. You'll pay more for furnished, though there are furnished options, just not as many.
If you want to live in an urbanización, talk to the huachimen. They know about places for rent. Go to the bodegas in your areas of interest as that's where people post For Rent notices. Talk to the shop keeper as well. Take a walk and look for For Rent signs in windows. If you want to live in a building, talk to the Potrero or anyone coming and going. Go to real estate offices. Many have listings online. We found our place by walking around and noticed a building that was just about finished with construction on the upper floors. Talked to the maestro who gave us the name and number of the owner. A week later the place was finished, we had a look, liked it, and moved in. Since you say you are flexible on cost and desire to learn Spanish, check out some language schools. Most also offer some type of housing, be it a home stay or furnished apartment - though again, you'll pay more than renting your own, unfurnished place. Arequipa is known as a good place to learn the language as it's spoken a bit slower and clearer than other places. Here are some other web pages where you'll find apartments for rent, though you might pay more. Some listings are private, some are from agents.http://www.adondevivir.com/departamento ... quipa.htmlhttp://ciudadarequipa.olx.com.pe/inmueb ... des-cat-16