Your general choice of calibers is interesting - -
Elder Son and I made two trips to Colorado for elk - - My choice was .338 and his was .300 WinMag.
Lest I "take on airs," I hasten to add that neither of us got an elk on either trip . . . . But we both like to visit with other hunters and shooters, in the field, in camp, and in the cafes. There are a LOT of hunters who take elk every year with .30-06 and .308 rifles with 165 to 180 gr loads. I know two Texans who use .270s with 150 gr. loads, and they don't complain of a lack of power, but they are both good hunters and good rifle shots.
I'd be willing to bet that a lot of elk fall to .30-30s in the hands of the locals, as well. I think the main thing is knowing the rifle and load, and choosing the shot critically. The hunter who travel far and at great expense to hunt, and has only a few days to fill his tag - - - Well, he might be forgiven for taking a shot under less-than-ideal conditions. The added confidence of the bigger .338 bullet, or the additional 75 to 100 yards range of the .30 magnum might be helpful to him, where the local might figure he'll have a better shot next week . . . .
jl98800 , no offense intended - - You style yourself a newbie - - Dunno if this means to large game hunting or to rifle shooting, or what. If it means rifle shooting, you might want to start out with a little less cartridge initially. Depending on the particular rifle chosen, both .300 and .338 WinMag can be a bit vigorous in the recoil department. Different shooters have different tolerances.
One suggestion for a newcomer to heavy rifles: Go to the gun club or a public shooting range and hang out a couple of weekends. If you can strike up a conversation with guys shooting some sizeable rifles, see if they have a few minutes to spare. Explain your interest and goal, and offer to pay for ammo to try a few rounds with various rifles. You may learn a LOT in just a few hours, and twenty bucks worth of ammo may save you a lot of expense in the long run.
An animal at the edge of the woods can be a little difficult to see clearly at 200 yards. It is not at all like a black bulls eye on white paper on the range. A good rifle with a good scope has far more intrinsic accuracy and energy than most hunters can ever properly use in the field. Being half out of breath, aiming at a walking elk from an unsteady position - - This is a far different proposition from being well rested and comfortable, shooting from sandbags on a solid bench rest. I personally feel that, as a responsible sportsman, I have no right to take a shot at a game animal if not certain of placing my bullet within about five inches of where I want it to hit. (A ten-inch circle at 200 years isn't all
that large.
In answer to that part of your question - - -
Always stipulating a familiar rifle with good scope, properly sighted in, under ideal conditions - -
.338 WinMag, 210 to 225 gr. bullets, out to 250 yards
.30-caliber magnum, your choice, 180 gr. bullets - - to 300 yards
.30-06, 165 gr. (some like 180, I know) - - 225 yards
.308, 165 gr. - - 200
.270, 150 gr. - - 200
.30-30, 150 - - 125; with 170 gr., to 100. If using iron sights, keep to WELL under 100 yards.
Above based on relative "power," ease of hitting with the given trajectory, and some very subjective factors. An expert rifleman with a lot of experience with the rifle/load, well familiar with the territory - - He can stretch the above somewhat.
Very best of luck to you.
Johnny