Since I killed my first elk back in 1966 and have killed one most years since then, I guess I qualify as one of those geezers with decades of experience. I can write about my experiences all day long, but I don't know how good I could be in telling a new hunter how to hunt an elk. I've lived and worked most of my adult life in elk country and have watched herds of 150 or more from my front window.
There is a load of good advice in the 30+ previous posts in this thread, and I don't necessarily disagree with some of it, I just don't do it. For example, I'll sometimes carry a print of a topo map or Forest Service land ownership may, but I have never carried a compass or GPS unit.
I guess my first piece of advice would be to be in good shape, both physically and mentally. I've seen elk in valley floors and everything in between up to timberline. You need to be able to walk over that next ridge and you have to want to walk over it.
Carry enough gear with you so that if you have to you can survive a night in the woods, but I have never hunted elk with my camp on my back.
Elk like open country, but they don't like people. They are primarily grazers and often will be out feeding in meadows at daybreak and sunset. If its hot or if they have been spooked, they will hold up in the thickest and darkest timber. A good tactic is to glass meadows and other openings, especially at first light, from a high vantage point. I have killed a number of elk by very slowly sneaking through the dark timber during the heat of the day. The scrub oak and aspen hillsides like is found in much of northeastern Utah and western Colorado are a favorite elk habitat.
Like in other hunting you should be very familiar and comfortable with your rifle. You should zero your rifle and verify where it is hitting at 200, 300 and maybe 400 yards off a rest on a bench, then practice, practice, and practice at those same ranges from field positions, prone, sitting, or resting the rifle on a bi or tripod or tree limb or backpack. I have killed my share of big bulls, and I am primarily a meat hunter now. I like the behind the shoulder, 1/3 up in the body shot. An elk's heart/lungs vital area is about a 15" diameter circle. We have steel gongs that size at our range, and I practice on them at least once a week for 2-3 months before hunting season opens. Head and neck shots can be deadly, but they are a very small target. I am not a fan of long range and Hail Mary shots. Elk are too magnificent an animal to let run off wounded. A gut shot elk can run for miles and will usually not be found.
Elk are not armor plated and any rifle in the .270 Win-7mm Rem mag-.30-06 class with a good hunting bullet will easily kill an elk, IF you put the bullet in the right place. I prefer slightly heavy for caliber bullets for elk. I have killed and have seen elk killed with cartridges below and above these, but I think that it is more important to shoot a rifle that you are comfortable with and are able to accurately placing your shots.
Undisturbed, elk like to walk on trails. One of my favorite elk hunting tactics is to slowly follow game trails. Walking on trails is also quieter than just blindly walking through the woods. Elk like to walk on the crest of ridges and you will often find their paths there.
When elk bed down in the woods, they tend to face downhill, looking for danger. However, it is easier to drag or carry your elk (parts) down to your vehicle.
For twenty some years I had my own horses that I used to pack my elk camps back into the woods and to carry my elk out. Dragging elk out with a horse will work, but the dragged elk likes to get stuck behind trees. I've had to carry a few elk out on my back, but I have never boned an elk out in the woods. It is VERY important to open up and dress out an elk as soon as possible, and preferably hang the quarters to cool. I usually don't skin the quarters until I process the meat as the skin helps to keep the meat clean and I won't have to cut the crust off the exposed meat. Putting the quarters in game sacks will help to quickly cool the meat and will keep it clean.
An un-gutted elk left on the ground overnight, even on snow, has a very good chance of souring and spoiling.
Don't forget to take a small camera with you on your hunt. I don't have pictures of my first dozen or so elk because I didn't carry a camera with me. Years ago I took a friend Shiras moose hunting near West Yellowstone, MT. Riding in on the horses in the dark the first morning, my friend whispered to me that he had forgotten his camera. So had I. At first light we came to a meadow with a bull moose in it. That moose turned out to be (at that time) the 4th highest B&C scoring moose ever shot in Montana. And we didn't get any field pictures of it. Also make sure that your camera batteries are fully charged. I have stories about that also...
So enough of my rambling and good luck, as luck has played a very big part in much of my hunting.