I fish carp for fun; one of the biggest fish you can catch easily. Once in a while I've taken them home to cook, just to see if they can be made palatable (they can, just takes some work).
In the spring spawn, you can use corn or worms or whatever. They aren't too picky. Try and fish an area which is shallow and flat and has little or no current. My favorite carp spawning spot is on the leeward side of an island in the Fox river. Last I was out was last weekend, and they weren't spawning yet, and they were concentrated in the deeper parts of the river.
Anyways, slow moving water is usually a good bet regardless of the time of year. Chum a few handfuls of corn in a spot which you know it'll fall on top of where you put your bait.
As for bait, I prefer hair rigs with corn on, but this is very problematic on rivers like the Chicago river with a ton of catfish and bullhead. Odds are you'll get more gut hooked bullhead than carp--you may want to switch to another bait, or a treble hook if that proves to be the case. Some people have suggested using Mcdonald's fries or Cherry tomatoes, but I haven't had any luck with these yet.
One guy on here I think said he gets into the really big carp in the Chicago harbors using roe from panfish he's caught. Basically, tying it up in a spawn bag. Makes sense, although I haven't tried it myself (yet).
Fox river, Chicago river are fantastic for Carp. Algonquin dam is dynamite if the water is low, fish under the part of the dam which has stopped flowing. The backflow part of the St. Charles dam is usually good for carp and catfish fishing a nightcrawler on a bottom rig, but I sometimes get smallmouth and walleye the same way there. Chain 'O Lakes I'm told is good too. Mazonia literally has a lake called "Carp Lake", but I haven't fished it.
I also walk the banks of a creek near me, because during the spring spawn big carp move up and down the banks of the creeks rooting around for worms. I have caught some beautiful fish just by sneaking up and plopping a worm in front of one.
If you're using a regular hook covered in corn, I recommend 6-10lb line, since visibility is a factor. If you're using a hair rig, it doesn't seem to matter at all. I have a heavy rod with 30lb. rod and I got to fishing hair rigs on it out of laziness. Works just fine; they aren't looking at the hook anyways, they're paying attention to the hair with bait.
Really though, it's hard to find a body of water in IL without carp upwards of 7-10 pounds in size.