I visited the GC January 2001 when I was road tripping around the country for 3 months. I think it was right around January 10th... New Years Eve I was in Tampa with friends and stayed a couple more days. Then I drove to Baton Rouge, hung out with a friend and we went to New Orleans and met some more friends. I was in N.O. the weekend after New Years. Then drove west 2 days to GC. Stayed in Flagstaff the day I arrived, then up to the GC, south rim. I only stayed 1 night in GC. Most of the support was closed for the season - limited lodging choices, limited food options, but it was open. Hiked both days, then out and drove to Las Vegas to see their craziness.
Hiked Bright Angel Trail to Plateau Point and back to the south rim the first day. That's just a little over 6 miles each way, pretty smooth footing. Just over 3000' elevation change, so very similar to what we see in New England. I remember most of the elevation change was in the first (and last) 3 miles, the middle 6 miles wasn't very steep at all. The elevation profile shows it gets steep between Plateau Point and the river and I didn't want to push it so I decided 6 miles down was about right.
2nd day I hiked Hermit Trail down to I think Breezy Point and back. That's around 5.5 miles one way, 2200' elevation change. Hermit Trail might not be open off-season any more, not sure. Hermit Trail was a lot rougher than Bright Angel, but still only what I'd rate as intermediate.
Both mornings there was about 2"-3" of snow at the rim, temps in the maybe high 30's to start, but warmed up on the way down. Plateau Point was around 60°. By the time I got back to the south rim, both days, the snow was almost all melted but not quite.
What I liked most is it was almost deserted. I expect that might not be the case anymore, but knowing how crowded the more popular national parks can get, it was a real pleasure to be able to visit one without many people around. The first day hiking, I think I passed about 20-25 people all day. The second day I saw a couple people near when I got started but otherwise didn't see anybody all day.
Hope to get back some day, and will try to go in the winter again.