12:59 a.m. on September 29, 2010 (EDT)
My first hike in the canyon was the whole month of October in 1983. First I went down for an overnighter, then a week, then two weeks coming out on Halloween. I went back every year after that and stayed from November to Feburary.
My 256 miler was in 1999, I went down three places prior and put in food and water caches about a week apart, so I would not have to carry it all or have to hike out to resupply.
I went down the South Bass trail in the west and came out the Tanner Trail.

Click on the image above for a better view. The red line shows my approx routes.
I went down the South Bass in the upper left and followed the Tonto Trail across to the Bright Angel Trail in lower right. Then went down to the river and up the North Kaibab to the North Rim. Then returned almost to Phantom Ranch and took the Clear Creek Trail to Clear Creek and back to Phantom Ranch.

This map shows my route to the North Rim and to Clear Creek and back to the Tonto after the South Kaibab. Then I continued the Tonto Trail to the Tanner Rapids area and took the Beamer Trail to the Little Colorado River and back to the Tanner Trail and out to Lipan Point on the Southeast Rim near Desert View. Click on the above map for a better view.
I actually did have to hike out to resupply on Horseshoe Mesa as someone had found and taken my last food cache. On the way out I met a guy who gave me a ride to the Grand Canyon Village ,waited while I bout 7 days of food and gave me a ride back to the Grandview Trailhead. I was back in the canyon within 3 hours of leaving it.
For a good book to read while in the canyon, get Colin Fletchers "The Man Who Walked Through Time" In about 1963 before the Glen Canyon Dam had been completed, he walked from Havasupai in the extreme western end of the canyon to the north rim on the eastern end near Marble Canyon, north of Desert View over 3 months. It is a very interesting read. He had to have 3 parachuted food caches dropped in as in the western section west of the South Bass there is few ways to get in bewteen Havasu and the Royal Arch Trails.
And/or else read at sometime, George Steck's books about hiking the canyon from 1945 to the 80s. He was a Math Professor at NAU (northern arizona university) here in Flagstaff and spent many summers,spring and fall breaks in the canyon. He is known as the father of the Grand Canyon.
Also read John Wesley Powells accounts of the canyon, he and his men went down the entire Colorado River in the late 1800s twice from Green River Utah to the mouth near present day Lake Havasu. He was a one armed General who lost his arm in the Civil War.
If you have time, stop into the Kolb Brothers photo studio on the South Rim near the Bright Angel Trail. They were the first two to make a lifetime of photographing the Colorado River starting in the early 1900s.
Personally I studied the Flora and Fauna and the Geology of the canyon during my 20 years from 1983 to 2003. I first saw the canyon in September 1983 while on a 8000 mile hitchhike around the USA at age 21. I vowed to myself that I would return ASAP to see the canyon, never knowing it would take me 20 years to continue exploring the canyon.
December and January are my favorite months in the canyon. When on the south rim its about 10-20 degrees and snowing and the inner canyon is 40-60 degrees, shorts and Tshirt weather after 30 summers in Jackson Hole, Wyoming. An average day along the Colorado any day of the year, is about the same as it is in Phoenix. In October 1983 I was still swimming in the Bright Angel Creek which flows off the North Rim.
I once went down in mid-July 1986, when my nephew came to visit while in Tucson and wanted to go in mid-summer. It was 119 degrees at the Bright Angel Camp and we and everyone else spent the day sitting in the Bright Angel Creek, then all day hiking out. I would never do that again!
Another great place to visit is Havasupai in the west. There the water is 70 degree's year round and has two beautiful high water falls.

Havasu Falls

Mooney Falls
And while you are in the canyon in October,I suggest hiking up fom the Bright Angel and Phantom Ranch to Ribbon Falls about 7 miles towards the North Rim on the North Kaibab Trail. See it below.

Notice the person just left of the falls for perspective of height?