Neil's right. I think your best bet will be following the Grand Illinois Trail. This is without knowing exactly where you're coming from or headed afterward or what you want to see in Chicago. If you're coming from the tri-state area (Iowa, Wisconsin, Illinois), the Trail starts not too far from Galena, Illinois (in the Northwest point of Ill.), so it may be worth picking it up there and taking it the whole way.
As far as finding your way around Chi, I find it best to utilize the city's route along Lake Michigan. In other words, stay on the Grand Illinois Trail. You could make your way through the city rather than stick on its outskirts on your way in, but it would take more time, and if you do not have much experience in heavy auto traffic, Chicago's streets might be intimidating.
The lake route makes it easy to navigate streets and keeps you away from them as much as is possible. Since everything's on a grid, you'll find Chicago to be easier to maneuver than most any other city you travel through. You can ride N or S and head W whenever you want to go in.
In my opinion, you'll be fine in this city so long as you're careful. Before you get here, try to know what it is that you want to see for those couple of days. There's a hostel downtown on Michigan ave., not far from the bike route; that may be worth looking into, though it's still expensive - something like 35-40 dollars pp. If you're with a group, you're probably better off getting a cheap hotel. If you're camping, try asking a fellow biker for their yard; you'll see a lot of them.
Anyway good luck. Here is another site that might be helpful: http://chicagobikes.org/bikemaps/