Pretty much anywhere will be within a 30 minute commute so you're good there. Apartments have been crazy hard to find so prices are up but you should be fine with a $1500 for a 1 bedroom. In fact I think you should try to find cheaper than that. A year ago my old landlord was showing my old apartment in the Armory, a nice 2 bedroom, for $1200.

The east side is nicer but more expensive and you might have a longer commute to Bryant versus the west side. Providence is a small city so while there are neighborhoods with more or less crime, it's kind of equal risk all around. Federal Hill and the Armory are affordable, close to rt 10 and 6 so you can avoid 95 (the main artery and source of traffic) depending on where you're going, and there's a lot you can walk to, especially restaurants and the Columbus Theater. Mount Pleasant is a little more sketchy.

The biggest thing you're going to want to figure out is parking. There's no overnight street parking in Providence unless you have a residential pass, and even then I believe you have to get the vehicle off the street for snow. It's a pain in the ass. If you find off street parking it's good to see what it actually is because I've lived places that cram you all in without assigned parking which can lead to neighbor conflicts, scratched cars, etc. One place I looked at expected you to leave your keys so that people could move your car to get out. Make sure it's a sane situation.

Hope that helps! And I'm sure others will chime in on anything I'm out of date on.

Edited to add: If you like Italian food, Federal Hill is heaven.