We love to camp at the Mogollon Rim! There are plenty of sites to camp in, but it does fill up quick on the weekends. By Sunday afternoon, almost everyone is gone.
When you first come in there are sites closer to the rim, but these tend to fill up faster than the others. If you drive further back into the woods, there are more options but no rim view. There are also hiking trails as well such as the Carr Lake trail, Military Sinkhole, Rim Top trail and a few more.
We camp off road 169; there are large group sites with a good amount of space in between. You can usually see the other people, but can't really hear them (unless they're loud campers). The road in is pretty good but does get rocky at some points if you turn off the main road 300.
I've taken a Hyundai Accent, but recommend a higher clearance vehicle if possible. Just go slow at certain points if you go down 169.
ALSO, please pick up after your stay!!!! One of our main sites was severely littered and it's sad to see that. I ended up with an entire bag full of other peoples' trash. Let's work together and maintain our beautiful forests. :)