For nature that's close to civilization, Deer Grove Forest Preserve is the best. While the camp sites are small, the amenities are super clean and the rentable cabins are as well. The trails through the grounds are varied and long--you can find something to suit just about any preference for suburban midwest hiking. There are plenty of paved trails as well as dirt, lots of fun little off-shoots into the woods, creeks run throughout, and there's even a little up-and-down thanks to ice age glaciers.
We stayed on tent-only site 009 which gave us a small view of the lake through the trees, a direct trail to the lake to sit with our morning coffee, and plenty of space between sites so we couldn't see or hear any neighbors. It's a short walk in to the site from the parking area in that section (maybe 50 ft total) and parking is easy and plentiful in the lot. We brought our own potty so I can't speak on the cleanliness of the facilities, but I can say they store their firewood indoors which was excellent for nice dry wood that was easy to start and burned hot.