No matter where you end up in the park, the views are amazing. The South Rim is the most popular, but also the most crowded. The North Rim is a longer drive, so less people overall. If there's no space available in the campgrounds at the north rim, try Jacob Lake area. You could also look to Fredonia or Kanab and daytrip to the North Rim.
My personal favorite place is Toroweep Point, but it's about 70 miles of dirt road out in the middle of nowhere. Very few people so there's always space, but you need the right vehicle to make it out there and I wouldn't even think about towing a trailer. More of an overlander experience.