I'd recommend the 6mm Remington (a 7x57 Mauser case necked down to 6mm or .24 caliber) for your "light" rifle (varmints, antelope, deer) and 7x57mm Mauser for your medium rifle for deer, elk, caribou, moose, hogs & black bear. Recoil is minimal with either round.
In addition to the 7x57, you could also consider either a .270 Winchester, .280 Remington or .30/06 Springfield as a medium rifle. I have listed these cartridges in the order of the amount of recoil generated by each using lighter to heavier big game bullets. Indicated approximate recoil does NOT include light-weight, varmint-type bullets.
The 7x57 is hard to beat since it seems to kill well far beyond the amount of recoil that it generates. Depending on bullet weight, recoil is about 15 ft/lbs to 18 ft/lbs using 120 gr.-175 gr.).
The .270 is another cartridge that seems to kill far beyond it's rather light recoil. Recoil about 17 to 19 ft/lbs using 110 gr-170 gr. bullets).
The .280 (7mm) Remington is a slightly better killer due to the heavier bullets it will "throw", but again... the heavier bullets tend to generate a bit more recoil. Recoil about 17 to 20 ft/lbs using 120 gr. to 175 gr. bullets).
Generally speaking, the .30/06 is a great round, but with 180 grain bullets or heavier (190-220 grain), it generates about all the recoil (approx. 19-26 ft/lbs using 125 gr. to 220 gr. bullets) a normal hunter can reasonably handle. Some hunters find the recoil of the 180 grain (or heavier) loads a bit much, but most hunters can handle up to and including the 180 grain loads without too much trouble.
Of course, the fit and shape of the rifle's stock and the overall weight of the rifle can and does affect the "sensed recoil" felt by the rifleman.
In addition, the ".308 family" of slightly shorter cartridges fall into these same categories... namely the .260 Remington, the 7mm-08 and the .308 Winchester.
I hope this information is useful to you.
Strength & Honor...
Ron T.