You may want to pick up a copy of "Photo Traveler's Guide to Arizona(323-660-8600; Or, write to them at P.O. Box39912, L.A. CA 90039). It gives great suggestions for photo opps at all of these locations & more. I went in the Fall. In winter you can expect the white snow to contrast dramatically with the red earth.
As for the Grand Cyn., you will only be able to get to lower elevation South Rim. The North Rim is closed. As for going to Antelope Cyn., I don't know if you can get in there in winter, or not. It used to be that you could simply drive in there with a 4WD vehicle. But, once it became popular, the Navajo's began to give "tours" (which amount to driving you about 2 miles through soft sand in the back of a pickup up to the entrance, and coming back for you in a couple of hours). When I went to the lower canyon, it cost $5 for the permit to enter Navajo territory, and an additional $12.50 for the Navajo families who own the land around there. The upper section of the canyon costs a little more. The Lower canyon is longer & deeper than the upper section. It is very challenging and requires ropes & ladders. I don't recomend this for the winter months. My guess is that it will be closed.
Monument Valley sits on a Indian land, as well. Much of it is off-limits, except by Indian-guided tour (for a fee, naturally). I don't know what time of day these tours run (or, if they run in the winter). If you have to go it on the public-access road, you should have a jeep or a truck. It is a not-so-good windy dirt road. There's half-day tours, 8-hour tours, etc. It depends on your interest in Indian ruins and artifacts. You will get some pretty spectacular photos even on the publi-access road, just not as close to the monoliths and spires. If you get above it, on a bluff and compress everything with a 400mm lens, you will get a very interesting shot. If you want to shoot the Navajo people, you need permision (and, a gratuity). Just keep in mind that much of the Valley cannot be seen (photographed) except on one of the tours.