Yep, that's a good ol' paint. Check with your state DNR before taking one home, but the way I read the IL regs, you may keep it if you have a valid IL fishing license. You must have a valid license for the entire time you keep the turtle; it counts against your possession limit.
Just a few of things to think about before you take a turtle home:
Baby turtles collected from the wild may be fragile and likely to die from stress-related infections soon after capture. Captive-bred turtles are likely to be healthier.
If your turtle does survive you may never re-release it to the wild; it is both illegal and irresponsible to do so.
Turtles are long-lived animals, so this may be quite a long-term responsibility.
Painted turtles are fairly small but still need a small pond or very large aquarium to thrive.