The Park gets nearly 5 million visitors a year, of which only 500,000 visit the North Rim. The South Rim has become congested enough on Summer weekends that the National Park Service is seriously considering prohibiting automobiles on the Rim, providing shuttle bus service from nearby parking lots instead. If you can visit in May or September on the North Rim, or any non-summer months on the South Rim, you will far fewer tourists. In July and August you will usually have a thunderstorm every afternoon.
Rainbows are common over the canyon.
The months of December, January and February are often cloudy with snow about once a week. However, in typical Southwestern style, the sun comes out immediately after a snowstorm and turns the Rim into a winter wonderland with crystal blue skies. There is no major carrier airline that services the Grand Canyon. If you are renting a car, there is a Budget office at the South Rim airport, and several car rental agencies in Flagstaff. However, I'd highly recommend renting in Phoenix and driving up.