I'm not sure about Il, but here in MI you don't need a license to cut grass. You only need one if you're doing Landscape construction....but not for maintenance.
As far as liability insurance, Safeco, Frankenmuth, and even some of the auto insurers have commercial liability. For part time, its not too expensive, I pay $164/Quarter. It's not "required" in order to be legal, and you can risk your entire savings......but its not advised. Back in College I was cutting lawns on the side, and one day I ran over a water access pipe in a new lawn I was cutitng. My mower cut right through the 1" pipe an sent it flying into the street. 2 houses down was a little girl riding her bike on the sidwalk. I bought insurance the next day for that very reason. I've never had to make a claim against it, but its good to know its there.
As for acquiring a tax ID number, it isn't necessary. I think its already been mentioned that you can set up a D.B.A. (Doing Business As). To get one you go to your county clerk's office, and apply for a business name you would like to use. It cost me $10. They give you a certificate, which you can then use to open a bank account. Now you can have people write check payable to your business name, instead of your personal name. But, this DBA uses your Social Security number for tax purposes, and therefore any income you make doing lawns becomes personal income, and you just claim it (minus expenses) on your tax return. You technically don't even need a DBA or a business name, you can always have them write checks to you personally, or cash. It's just a matter of preference as to if you want a "company name."
This is a great forum for questions.....I've learned more on here than anywhere else!
Best of luck to you.