I am looking for a first rifle that will handle deer, moose, and elk. I have hunted with other guns in the past but now I want one of my own. Looking for a caliber that can reach out to 300-400 yards. I have done a fair bit of comparison on ballistics and the .270 and 7mm Rem Mag seem like some of the flattest shooting with very good ballistic coefficients. Also want to avoid the kick of some of the larger magnum calibers like the 300 win mag, 338 win mag, etc.
You may, or may not, realize you just asked the equivalent to "What's the best oil to use in my new motorcycle". Which is what spawned the following ten pages or so.
All of those calibers have been reliably killing those species of game since before most members here managed to find their way out of the birth canal. And the perfectly suitable 30/06, unmentioned, for even longer than that. And all of them can certainly reach out to your mentioned 400 yard range. Assuming you can actually reliably hit the vital zone of a game animal at 400 yards, from the various field shooting positions, rather than just blow a jaw or leg off here and there.
My personal favorite is a 30 Newton, built on one of Husqavarna's featherweight actions. It is also a caliber as rare as rocking horse poop, factory ammunition hasn't been made since before WWII, and does little that a 308 Norma or 300 Win Mag also won't do, except it doesn't have a belt nor the secret magic word "magnum". So realistically, in the real world, if I had to hunt the rest of my days with a 30/06 or .270, I would lose little. Raging debates over ballistic coefficients, G1 or G7 models for BCs, etc are great sport, but they don't drop game.
As far as recoil goes, it is determined by the weight of the firearm, the ejecta (weight of the powder charge), projectile weight, and muzzle velocity. The chambering and/or bore size does not figure into the equation.
You can't cheat the laws of physics, other than to ensure the rifle stock fits you well. If you're not used to recoil, you can have a different conversation regarding how to deal with it with lighter handloads and so on. If you're just starting, why NOT a .270, .308 or 30/06? There's no law against selling them to buy a bigger chambering later on if you decide you're losing game because they're inadequate for the opportunities you get in the field. What is expensive is curing flinching and other bad habits by starting higher up on the recoil food chain. My wife's first centerfire was a .243 Winchester; elk and deer obediently died when she pointed it at them. She's now quite comfortable with a .358 Winchester with Barnes 180 grainers built on a pretty light rifle, but the game doesn't get any deader because the bullet is heavier.
So I'll suggest you ask friends/family to allow you to shoot their hunting rifles in assorted calibers to get a feel for them. If nothing else, you'll get a feel for what the recoil for various chamberings in rifles of similar weight is like. Another consideration is how pleasing the rifle is while in your hands, being carried, etc. My featherweight Husqvarna doesn't hang as nicely on the target while standing or kneeling as my much heavier Sako does - but when 99.9% of the time in the field it's being carried and not fired, the Sako really doesn't get out much, despite posting slightly smaller groups.
Then if you're still unsure, buy a .308 or 30/06, some reloading equipment so shooting is more affordable, and teach yourself/learn to shoot accurately from field positions to give yourself an idea of whether or not you can ethically shoot at game at longer ranges. Any game animal you hit with either of those calibers in the vital area will obediently die, ballistic coefficients won't matter the slightest. As far as that goes, you'll get pretty similar results with a .270 Winchester.
