The cost of living in Tucson is generally relatively low, which proves favorable when searching for an apartment or house to rent. The typical rent for a 1-bedroom apartment in Tucson is roughly 500/month, though certain areas and apartment complexes will be lower or higher. Rent costs can also be reduced by living in multi-bedroom apartments or condos and having roommates.
Perhaps the best advice for those looking for housing is to contact the current graduate students and see if they know of any available rooms, apartments, or houses, or to get their advice on which neighborhoods to live in or avoid. It may also be valuable to do research online using Craigslist, apartmentguide.com, or apartmentratings.com to find a suitable place that you are looking for. Do not be afraid to ask if you can stay with a graduate student for a few days when you first arrive here so that you can find a place!
Another great resource is the University of Arizona's Off-Campus Housing Guide, which contains a search engine listing available housing around campus and around town, and a roommate search.
The UofA has dedicated on-campus apartments for graduate students in a complex called La Aldea. This gated community has 1-, 2- or 4-bedroom units available. They are more expensive than comparable rental properties in Tucson due to their location. It is not a super common option given the availability of apartments and houses in Tucson, but those who have lived there say it's nice. You can also look in the Daily Wildcat for apartment listings, or really, just talk to the graduate students and we can help you find a place that suits your needs. There are plenty of great websites to help with this, too. The Tucson housing market is quite affordable compared to most other cities.
Other resources you could try:
ABODO They apparently also offer a free service called Search For Me that will go and find rental options based on a student's needs so you don't have to look yourself.