Old Cane-
For me, medium and big game rifles start at 6.5mm/.264" and go up. The 6.5x55mm and the 260 Remington are two of the easiest recoiling, and they're also the easiest to find factory ammo for. My local WalMart stocks both. These two rounds are ballistic twins, and there's no logical reason to pick one over the other, other than finding a particular gun that fits you. If you reload, commonly available bullet weights start around 80 grains and go up to 160. The 100 grain Nosler Partition is a good deer bullet for a 26 caliber. In my tests, these bullets penetrated an average of 19.5" of saturated newspaper at 200 yards, with an average retained weight of 72.23 grains. Average expanded diameter was .536". All were fired from a 24" long 260 Remington barrel, so velocity at the muzzle was just a hair over 3300 fps. I didn't chrony the impact velocities, but the books claim it should have been around 2700 fps. That's not bad for a light-kicking deer rifle, and you can move up to heavier bullets for heavier game, which will make the kick feel more like a 308 Winchester. Stay away from 140 grain Nosler Partitions in the smaller-cased 6.5mm rounds, as I've found that they don't expand well at 200 yards when fired from anything less than a 6.5-06. Uneven expansion causes the bullets to tumble, losing the rear core and giving relatively shallow penetration. Since this happened repeatedly in testing, I've opted not to try it on any animals. All the other bullets that I've used worked just fine, both in tests, and a few critters that I've had the opportunity to harvest with 6.5mm bullets.
If my math is right on a table that I drew up, an 8 lb. rifle/scope combo chambered for the 260 Remington, firing a 120 grain bullet, will have 13.3 ft. lbs. of recoil energy. To have the same sectional density and penetration potential, a 30-06 would have to fire a 165 grain bullet, giving 22.94 ft. lbs. of recoil from the same weight rifle. Regardless of stock design, your shoulder will notice the difference. If you go after elk with your 260 with a 140 grain Speer grand-slam, recoil energy will still only be about 14.1 ft. lbs. Rifles in the 26 and 27 calibers are a little small for elk, IMHO, but a lot of them fall to these calibers every year (along with plenty of moose, especially in northern europe).
You might also lean towards using a heavier rifle, since you sound like a sturdy fellow, and then your rifle's inertia can help you out.