On the South rim...
Mather is like a zoo in the Summer, however, it might be OK this time of year. You can check for reservations and see if you can find a spot without neighbors. Once you arrive if you find that you have neighbors, you can go back to the campground entrance to see if there are other campsites available that might be open & better for you. You can set-up once and walk to the shuttle bus stops pretty easily. Our neighbors were about three spaces away from us on one side and the other side was about 7 spaces away. There is a significant amount of night time road noise from everywhere in Mather. They do have a shower house. Elk & deer walk the area at night.
Outside the park, south of Tusayan by 2.5 miles and south of the RIM by about 8 miles is "Ten X" campground that is very nice. You'll be driving into the park anytime you want to go there, however, the campground hosts that I experienced there last Fall had a little sign that they put on my fee tag post that said I was out exploring, so no one took my space. The place was close to Hwy 64, however, we did not hear any night time traffic sounds like at Mather. The campground was about 1/3 full last Fall.
On the North Rim...
Higher elevation than the South Rim, so watch for weather... snow, etc., and the whole area might even still be closed until May. So, for future reference...
I've never been able to stay at the North Rim campground, it's always full, however, reservations are possible. Totally less zoo like on the North Rim. And, from the campground you can walk everywhere.
From Jacob Lake (gas available, junction of Hwy 89A & 67) it is about 20 miles as the crow flies going south to DeMotte Campground, outside the park entrance, Sort of like Ten X, however, smaller and not quite as nice. It's about another 15 miles to the North Rim.
Just south of DeMotte, is a west turn on Road 22, near Deer Lake (a small pond or depression when no water is present) that will take you about 8-9 miles on the dirt to the DRY PARK meadow and dispersed/free camping. Road 22 is marked as one way to get to Freedonia, all dirt, about 50 miles (better later in the year when there is little rain or snow, if it's even open).
The road to Cape Royal and Point Imperial are not to be missed, however, like the rest of the area will probably be closed until May. No camping near these areas, it's all inside the park.
That's it from me.