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Your criteria suggests the East Side of Providence as being your best location since it is closest to Brown. Rent for a decent 2 bed will start at about $1200US plus utilities (heat, electricity, cable). A separate house ranges $1600-2500. Best not to smoke or have pets since that can complicate your ability to rent.
If that is too expensive, you'll need to look farther away- here's the bus map for the RIPTA:
RIPTA: Rhode Island System Map (http://www.ripta.com/content1306.html - broken link)
Neighborhoods for you might be Barrington, East Providence (Riverside or Rumford), Edgewood in Cranston, Elmhurst in Providence, Pawtuxet Village in Warwick, Oak Hill in Pawtucket.
I believe the question was about the city of providence not the state of R.I. ...........these people make it sound like providence sucks big time so live in another smaller city. PEOPLE, if you aren't answering the question at hand then DON'T ANSWER.