Yeah, safety is definitely important. That was a big thing for us as well. From what I saw, if you're looking at places right around the loop, they're probably going to be pretty small! We were looking at 2 bed places with a budget of $1200 (rent only), so I couldn't really tell you much about 1 bed rents. It seemed like most areas of Lincoln Park were pretty good, but a little pricey. Wicker Park or Bucktown might be decent areas for you to look at if you're trying to stay close to downtown. If you don't mind a bit of a commute (like 1 hr to the school), Edgewater and Andersonville are both great, safe neighborhoods. We mostly looked at places on the northern end because I really don't mind the commute and we wanted somewhere safe with a bit of green space to walk our dogs. Like I mentioned above, we ended up choosing a place in North Center. It's very safe, but a bigger commute than the loop neighborhoods (40ish minutes on the El to UIC). Logan Square is nice as it's a very short commute on the blue line to UIC, but it's the hipster neighborhood (which is fine if you fit in with that crowd) and parts of it touch some rough areas near Humboldt.
Apartment People and Apartment Finders were both useful for seeing the neighborhoods, but I didn't love most of the apartments they showed us. I found Craiglist and resources such as the Chicago Reader or RadPad to be more useful. It also works pretty well to drive around the neighborhoods you like and just call numbers on For Rent signs that you see.