The Grand Teton National Park is open all year long with the exception of Christmas day. The only available places available to stay inside of the park in October are camping sites. I would recommend staying within the nearby town of Jackson. Travelocity has a lot of great deals for hotels in Jackson. You can search their website by going to: http://www.travelocity.com.
Supai village, located within Havasu Canyon, a large tributary on the south side of the Colorado River, is not accessible by road. The Havasupai Tribe administers the land, which lies outside the boundary and jurisdiction of Grand Canyon National Park.
The trail to Supai begins at Hualapai Hilltop, 191 miles/308 km from Grand Canyon Village or 66 miles/106 km from Peach Springs, Arizona. Peach Springs has the nearest services – gas, food, and water. No services are available at Hualapai Hilltop.
More information Supai and Havasu Canyon is available on nps.gov.
You must absolutely follow highway 87 (Needles Highway) to Mt. Rushmore. Out of Hot Springs, drive about 11.5 miles North on 385 , until you get to 87 North. It will be easy to follow from here to Rushmore.
This road is one of my favorites that I have driven in South Dakota. It winds through the mountains on narrow roads, but driving slowly and stopping often, you will be able to stop at multiple lookouts and see the monument from afar. There are also tunnels through sheer rock faces, and all together a fantastic drive!
The USGS administers the Golden Eagle Pass, and will have a complete list of where they are sold. You can go online and order one at www.parkpass.net, or get in touch with USGS ordering line directly at 888-275-8747, ext. 1, or email them at firstname.lastname@example.org
Enjoy your park travels, in Illinois and anywhere else you go!
Acadia National Park in fact does not have any structured lodging within its boundaries. That said, one of the things that makes the park (and the towns on Mount Desert Island) so special, is the interconnectedness between the community, its resources and the park itself.
The Bar Harbor Chamber of Commerce is the absolute best resource for ALL things Acadia.
My favorite times to visit Yellowstone is in May/June and September. There aren't nearly as many people as there are in July and August. The weather can be unpredictable during the spring and fall, and the wildlife is amazing. In the spring, there are animal babies, and in the fall is the elk rut. Check out http://www.nps.gov/yell.
I agree US-20 would be the best but a tad longer.