Ah, Ligonier -- cute little town!
Suburban Chicago is vast, and not very interesting. Going around means going WAY around.
One option is to head into Michigan and take one of several ferry options across Lake Michigan into Wisconsin. Here's
a list of your options.
Another option is to slow down and enjoy going straight through the center of the city. Just be an urban biker with panniers attached. It's fun! I rode through New Orleans this way on St. Joseph's Day (the biggest parade day outside of Mardi Gras) and it was one of the greatest touring days I've ever experienced. I had no idea what I was biking into -- just figured they'd heard I was arriving and wanted to throw a party to welcome me!
If you don't have a mileage goal and are willing to improvise your route, you'll have a terrific time in Chicago. It's a great city, and a bike is a great place to see it from.