You would have to get a back country permit in order to do that. Parking is usually not an issue as there are 4 lots next to the visitors center. If you want they also have a year national park card for all national parks for about $80 this will get you in for free at all national parks. You can hike down for free but in order to camp you need a permit. Calling the backcountry services will help you on regards to any questions you may have.
Hiking to the bottom of the canyon is free. You have to pay for your car (I paid $35 for compact) to buy 7 days ticket when you come to Grand Canyon National Park. There are a lot of places to park your car, it will be no problem.
On the South rim I saw hikers coming out of the canyon with alot of gear. Check out somewhere close to Grand Canyon Village.
Contact visitors information and they will guide you You can park And they do have overnight tent camping for about$40 p.p
Contact the Grand Canyon Backcountry visitors center for the most accurate information. Bright Angel trail is about an 8 or 9 hour day hike from the South rim.
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