The .270 shoots flatter than the others and with the right bullets, will kill moose, elk, and bear, although I think the .300 Weatherby is a wise minimum for huge brown and polar bears. The 340 Weatherby is probably a really good polar bear rifle, with enough velocity to hold up well to the velocity loss imposed by extreme cold.
The .243 is mainly just a varmint and deer rifle, if the deer aren't too big or very far away. The .270 shoots flatter than the .308 and most .30/06 ammo, and is a little easier on your shoulder.
I'd buy ammo loaded with Nosler Partition bullets for larger animals. I even like their Federal Premium/Nosler load as a basic .270 round, as it leaves you with a margin of adequate bullet integrity if you have to shoot a bear while deer hunting.
The .338 or .375 H&H Magnum (a really GOOD ctg. for larger animals) might be better for hunting Roosevelt elk up there in heavy timber. But those rounds recoil a lot more, and if you have to ask a question like this, you are no rifle shooter! The .270, especially with the Nosler 150 grain bullet, will kill elk well if you know their anatomy and can place your bullet well at any reasonable range.
The famous hunting writer and gun authority Jack O'Connor (1902-1978) preferred the .270 to any similar cartridge and he killed a LOT of animals with it, including moose. He felt that moose are huge and take awhile to die, and those he killed with a .270 died about as quickly as those shot with a .375 H&H!
When I lived in Newfoundland, the local hunters mostly used .303 sporters, the .308, or the .300 Savage, in one case. None had any trouble collecting moose. Some had .30-30's. BTW, moose tastes more like beef than whitetailed or mule deer does, and is heavier meat with more fat. I liked it.
O'Connor killed one elk at 600 yards, using a .270 and a handload with the 130 grain Nosler. I think that was extreme range for ANY hunting rifle. He usually took lung shots. That elk was lying down and Jack had a rest to shoot from.
I was curious one day in Newfoundland and was shooting in a big pit on base. I had a Husqvarna .270 and a SMLE .303. I fired the .270 into a big bank of fairly soft mud. Now I know that this was not a scientific test of bullets as in controlled firing into big blocks of ballistic gelatin. I forget whether I was using CIL/Dominion Sabre tip ammo or Winchester Silvertips. But the bullet made a SIGNIFICANT cavity in that mud! I quit worrying whether it'd kill a deer...or a moose!
If your library keeps old issues of, "Outdoor Life" with O'Connor's gun columns or you can buy his gun books, get them and read his stuff on the .270. It will fill you with confidence, provided that you can shoot well. He owned .30-06, .338, .375 H&H , and .416 Rigby rifles. But even in Africa, he and his wife usually used .270, 7X57mm Mauser, and 7mm Remington Magnum rifles on most game. As I recall, Jack saw little practical difference between the .270 and the 7mmRM. He felt that a .270 with 22-inch bbl. that weighed about eight pounds was both effective and easy to carry, compared to heavier rifles that needed longer barrels with Magnum ammo. And it kicked less.
He favored the .257 Roberts as a varmint and deer load, over the 6mm rounds like the .243. But he knew that the .257 needed good handloads to reach its potential, as the factory loaded it a bit light, with RN bullets that weren't suited to retain velocity well at extended yardages.
Anyone who likes rifles owes it to himself to own O'Connor's "The Hunting Rifle" and, "The Rifle Book." John Taylor's, "African Rifles and Cartridges" is the other rifle book that I strongly recommend. Much there translates well to US hunting conditions. But modern bullets are better.
The Ruger M-77 is a very good rifle, if the bottom of the stock around the magazine floor plate is a bit square for my taste. You might also look at Winchester M-70's and the CZ rifles. My favorite is the M-70 Featherweight Classic. The Sako is also very good, if a bit heavier. It lacks the controlled-round feeding of the others, but is usually very reliable. However, I had one on which the receiver turned a plum color. But I suspect that can happen with other brands, too. The Sako was otherwise very impressive. It was a Finnbear, made in the early 1970's.
If your budget is limited, I understand that the Savage bolt actions are excellent values and just plain good rifles. I haven't seen any recent ones, but that's probably a distributor/dealer thing and a result of customer demand here, where the only Savage that seems well known was the M-99 lever gun. And most younger hunters don't even know it!
I'll have to look up the Ruger American, but the M-77 in general is a sound rifle. I think it can still be ordered with iron sights, too, a major boon.
Been wanting one for a month now and I think I got the last one available within a 500 mile range. Now to find the right scope for under $100.00. 
