I have been photographing bull elk for 7 years. Along the way I have made most possible mistakes and learned to overcome them. The following tips should help many take much better elk pictures.
1. Most photographers are after pictures of big bull elk. They must always keep in mind that during the rut, bulls are full of testosterone and can get aggressive if you crowd them. Learn to read their body language when they feel challenged -- they will square up on you, they literally get their back up, and if you look close you can see their nostrils flare. If you create this situation back off some. This is why god created telephoto lenses. If it feels tense to you, it probably is. Trust your instinct. You will also get better pictures if you and the animal are both at ease. The picture with this entry is the Boxley Stud, a monster bull. I took it on my belly looking up at him pretty close -- I would say it was 30 feet -- too close. He is one of the dominant bulls in Boxley Valley near Ponca, AR.
2. Elk as a subject. Elk are light on top, and have dark necks and bellys. I am very aware of exposure to avoid blocking up the dark areas of their bodies. Generally it is better to set up your camera for spot metering and perhaps to over-expose 1/3rd of a stop. I always shoot in RAW because it gives me more room for error.
3. Elk as a subject II. Think of an elk in terms of its component shapes. The body as a rectangle, the Rack as a hoop, the legs as sticks or lines. I believe the best pictures come from breaking the straight lines, through body twists and turns, and legs crossing or bent at the knees. Straight on side shots look like box cars or cattle. There are reasons why sculptors show curves and twisting in their works. Think about the statuary at Bass Pro and Cabellas -- you will get the idea. Break up those straight lines.
4, Equipment. My standard rig is no tripod and my DSLR with a 100-400 zoom lens, I usually try to shoot at ISO 2000, aperture priority, +1/3 stop and at least f8.
5. Composition I go for "a day in the life of" pictures, I want a real or implied story. Also I prefer movement or two animals interacting. A good compo pulls people in. It makes people guess what the animal is feeling. It has an emotional aspect to it. It compels participation by the observer. I also love color and long tonal scales. Something special about the light. You have to observe and think to get great compos. Also pre-visualize away from the camera. Imagine your "bucket list".