Illinois taxes are extremely straightforward, even for part-year residents (as I will be this year!). You can see the form here, or even better file electronically on their website for free.
There are very few exemptions; the main one is for dependents, documented here, $2375 this year. There is no rent deduction, though there is a credit for property tax if you own; also money paid for school or saved for school, and a few other small things (see here for a partial list). For the most part, though, residents pay 4.95% of their Federal AGI minus their personal exemption.
Since you're a part-year resident, you will fill out a form specifically for that which will determine your portion of income owed IL taxes on, and exemptions. See Schedule NR, and Schedule NR instructions. You'll end up multiplying your exemptions by the proportion of your income that's attributed to IL, so if it's 1/4 (sept-dec and you had same income all year) you'll get 25% of the $2375. But again, try to do this online if you can - it's free and quite easy (though I've not done Schedule NR before, so can't promise that is as easy given how many lines it has!)