Hello, fellow Chicagoan! In terms of affordable places to live, there are a ton--Chicago is one of those places where you'll find a huge range of rent prices, so you'll definitely be able to find what you need. Lincoln park is where a ton of young professionals live, is incredibly safe, but may be a little pricey (although I have a friend living there for $700/month, so if you look hard enough, you can find something affordable! Especially cuz Depaul University is there so a ton of apartments cater to student prices.) Lakeview, Logan Square (stay close to the El stop though), Andersonville, and parts of Wicker Park are good places for young people too, that are safe and definitely public transit accessible. If you want to be adventurous, you could live in Hyde Park for cheaper than those. The neighborhood Hyde Park itself is safe, since it's where University of Chicago is located, but be wary of taking public transit at night because it borders some unsafe areas of Chicago on the South side. You'll get cheaper rents for that reason though, in the $500-600 range. If you stick to the other neighborhoods on the North side of the city, you should be able to find rents in the $700-900 range. As for finding roommates, try Craigslist! I'm sure some universities in Chicago have grad students looking for roommates too, so if you try looking on Depaul/Northwestern/UChicago/Loyola's housing sites (I'm assuming they have some) you'll be bound to find someone.
Most importantly, invest in a good coat, it'll be a cold and long winter! Let me know if you have any other questions!