Is there a fee to enter the park?

No, the park is a non-fee area.  Donations are accepted and can be made at the visitor center.

Do I need a guide or backcountry permit for the North or South Rim Drives?

No, the North and South Rim Drives plus the White House Trail are self-guided activities you can do without a guide or backcountry permit.  The rim drives and trail are open all year long.

Do I have to make reservations for the campground?

No, the Cottonwood Campground is open all year on a first-come, first-serve basis and is a non-fee facility.  Keep in mind there is limited services during the winter.