There are a lot of beautiful sand dunes and forests along the shores of Lake Michigan in Indiana and Michigan. You can actually take commuter rail from Chicago directly to the Indiana Dunes National Shoreline. If you have more time / money and want to put in a little more effort, try going farther north (you would have to take an Amtrak train or drive) to one of the many towns along Lake Michigan. Pretty much every major town has a huge state park close to it, and there are some really cool and beautiful landscapes. Try taking the route towards Grand Rapids and stopping in Holland or New Buffalo. Or you could take the train to Kalamazoo and hike to South Haven (a town on Lake Michigan) along the trail, If you have access to a car, try going even further to either Silver Lake State Park, Ludington State Park, or Sleeping Bear Dunes in "up north" Michigan. There are also long trails that traverse the state going both north-south and east west as well as more rustic national forest areas that have longer trails that are farther to the north. Enjoy your trip! Chicago is amazing in the summer, make sure to take pictures at "the bean"!
Here is a website of trails in Michigan...all different types, although nothing like types out West or along the Appalachian Trail.