Guns in general: You get what you pay for. Sometimes you get a gem, sometimes a lemon. If you can, try before you buy. You didn't mention budget, so some of this might be unreasonable.

Any manufactuer names or model numbers I mention are individuals or companies I have no financial interest in, nor am I associated in any way with, other than as a satisfied customer in certain cases.

Every gun person has opinions, and some have knowledge. Here's some of mine. It is up to you figure out where or if it fits in your world.

I've know little 5'2" females shoot 44 magnum handguns and 375 medium bore rifles just fine, and 6'2" big framed men gun shy with a 12 gauge. I submit a good shooter needs to shoot more than 3 rounds the weekend before hunting season. 300 over a couple month span, maybe. Learning the proper way to hold a weapon, and how to properly control the recoil goes a long way to what an individual can tolerate. With a hand bedded action and free floating barrel, most rifles shoot better than the person behind them can hold.

Here's something you might have never considered: snares might be more effective that any firearm, pound for pound. Legal or not, in a real survival situation, I'd take snares over a firearm. It's an effective force multiplier. websearch buckshot snares for pre-made ones.

The distances I discuss below are what I personally would feel comfortable taking a deer sized animal at, based upon my knowledge of ballistics, the un-customised rifle models or shotguns I have used in those calibers, and factory ammunition.
In some cases I may know enough about a load to be able to 'lob' a bullet because I may have shot hundreds of rounds of that particular load to know how it will perform in MY rifle.

Your results will vary.

Shot placement is more important at normal hunting ranges (under 500 yards) than bullet weight, velocity, or anything else. If you hit the target and penetrate into the heart, brain, spine, even with a non-expanding solid, it's down. Lung shots often entail tracking the animal for awhile.

Having said that, are you in the market for an off the shelf weapon, or a customized one ?

Can you legally use a rifle in your state for hunting (whatever ?)

A 12 gauge has a lot going for it, close in. With a slug barrel and scope, maybe 175 yards max. change the barrel and its good for birds. 12 gauge is kind of a universal weapon, ammo is inexpensive compared to rifle ammo, in a survival situation there are even flares available. Knoxx Compstock goes a long way on recoil reduction. Many manufacturers out there. I recommend a pump over anything else.
bird or slug barrels are relatively cheap and easy to interchange.

if you can use a rifle,
reasonable odds are it will see a lot more day hunting trips than expedition hunting. Shaving weight off a rifle increases the price exponentially both from a materials and manufacturing point of view, and a decreasing market share point of view.

I'm not going to discuss the ethics of taking game at long distances, if you, the shooter has not had enough experience firing at the distance the game is at, don't send that round.

when you properly fit a stock, glass bed the action, free float the barrel, you increase the distance the rifle is theoretically accurate at by hopefully mitigating stressors on the barrel, and the human behind it.

Action design:
Leverguns are not possible to really freefloat the barrel or bed the action, therefore they may have some accuracy problems at long distances due to barrel harmonics. Changing wood to Kevlar helps.
Boltguns can have the actions bedded and barrel freefloated.
Both can usually have muzzlebrakes added, and the stock properly fit and a good recoil pad added. There are two common bolt gun designs out there, the mauser type controlled round feed, and the remington push feed. both are good. If you were hunting dangerous game, the mauser type is preferred by most guides/PH's who don't use a double rifle. Most american sniper rifles use
actions based upon the Remington action to some degree.

Cartridges:
243 Win is a classic eastern whitetail cartridge. Good out to about 450 yards if the person can hold it.
It's never been a military cartridge, so it can travel to places where civilian posession of military cartridges are banned (africa, anyone?) Doesn't kick too bad. disadvantages, ammo is expensive, and barrel life is short. I wouldn't hunt medium skinned game with it. Deer, antelope OK, elk or bigger, No.

308/7.62x51 NATO, militaries have and continue to use this cartridge, so there are some countries you cannot take it hunting. Advantages include a large selection of factory loaded ammo, from 55 grain sabots to 220 grain thumpers, with 155 grain and 168 grain being 'standard' and 173 being typical 'match' loadings. Bullet selection is important, and everything from varmints to brown bear has been killed with a 308. Surplus ammunition is available. Nothing makes me warm and fuzzy like a sealed case, just in case. Reloading components are easy to obtain. Recoil is manageable, but: restocking the action to one that fits correctly, addition of a good recoil pad, and adding a muzzlebrake goes a long way to adapting something to cover both intentions. Some mass manufacturers sell them with a muzzlebrake, bedding the action and freefloating the barrel gets the most bang for the buck if you think you're going for a long shot.
Very few cartridges are as versitile as a 308, either. The Palma matches are shot at 800,900,1000 yards with the 308 cartridge and open sights. For an off the shelf one that can be rapidly adapted to an individual shooter, look at Tikka. Some models have a spacer adjustable stock, and have a threaded muzzle for a muzzle brake.

7-08 is an attempt to get more velocity out of a 308 by going to a smaller bullet, moving faster, with higher B.C. bullets at the same mass. It's a semi-wildcat cartridge, and regardless of how many manufacturers offer a chambering, it's still not likely to ever be available at a resonable price in bulk compared to a 308. I do not see the slight advantage of velocity and B.C. outweighing the advantages of the versatility of a 30 caliber tube. I often refer such former wildcats as a solution in search of a problem. very few were more sucessful than the parent cartridge. that said, here's a few notable exceptions.
30-03/30-06 I think #1 in number of US manufactured rifles chambered until the 5.56mm overtook it somewhere in the 1980s ?
22-250 savage, 220 russian/22ppc,6mmPPC, 6.5x284 winchester. used for varmint, benchrest, and long distance, respectively.

With all due respect to benjamin, in his post, I do not agree with a choice of a 7-08. It has a lot of negatives, and the positives it does have can be satisfied by relatively inexpensive modifications to rifles with much better capabilities and ammo availability.

30-06, parent case of many, many other cartridges. Still sometimes considered a military cartridge, sometimes not. Comparable to 7.62x54, 303 british.
Classic deer cartridge, enough to kill anything in north america out to 500 yards short of moose or bear, with heavy loads, those are harvestable at shorter ranges as well. It's kind of fell off the wayside with the gunrag hyped ubersuperduper mags of late. recommend a good brake, bedding the action and freefloating the barrel. done right, it's like a 243 in recoil.
ammo is readily available and surplus can be found for practice.

30-30, legacy of a bygone era, with a thin metal buttplate, mine kicks like a mule. Replace the wood and metal with kevlar and a recoil pad, easy to shoot. Not good beyond about 300 yards, and I hesitate beyond 200. beware of top ejectors and scope use.

45-70, another legacy. but.. with good semi-factory loads, and a muzzlebraked gun..no worse than a 308, and will take anything in the world with proper loads. examine the Marlin 1895 guide gun. reach out to maybe 300 on non dangerous game.
commonly seen where there be bears about.

compact takedown 45-70 levergun: look up Jim West, Wild West Guns in Anchorage, Alaska. called the Co-pilot, for a very good reason. He also does very lightweight expedition rifles.

Levergun with pointy bullets. Winchester 1895 has been
chambered in 7.62x54R,303 british, 30-06,30-40, and 405win.
here's a levergun that you can shoot pointy bullets in, as it doesn't have a tubular magazine. 405 has taken every big game animal on earth. doesn't kick awfully bad in 405. 350 yard gun in 405, longer on 30-06.
You can find them available in take down, and scope mounts are available. One of those with a good brake on it might be a good choice for an all around levergun, particularly in 30-06. Hang a good 3-9x scope on it (Sightron, on the low end, Nightforce on the high end) and go hunting.

That being said, if you're not in the market for a high end gun, a remington 700 from wal mart isn't a bad choice if you keep it under 300 yards in a 243 or 308 chambering. I would however, go with a good scope. Sightron is a good one for the money.
avoid cheap scopes. period.
A Marlin 1895 in 30-30 or the guide gun in 45-70 with the brake are also good choices. Mossberg 500 slug gun, I'd put a knoxx compstock on it before firing heavy slug loads.

for some drool fuel, at the high end, Ed brown, wild west guns ultralightweight expedition rifles,Or WWG co pilot in 457 ww magnum. Take down bolt gun, HS precision. Then price a nightforce scope.

whatever you get, shoot it enough with your chosen hunting ammunition so you know how it performs at your hunting distances.

I carry salt, pepper, tobasco, a firesteel, some snares, and an early Fairchild/Armalite AR-7 with a box of 22 stingers in my day pack. In the unlikely event I find myself in a surivial situation I think I might eat better, longer that way. My hunting rifle chambering is always sized to take care of the biggest problem animal I might encounter, and it depends on where I'm going, what I'm hunting, and how far I plan on walking.

I am in no way advocating snaring (or poaching, hunting out of season, and so on) as a way to procure game animals for anything other than a genuine survival situation.