buffybr

Like others have posted, a rifle is very superior to a shotgun for deer hunting -- if rifles are legal where you hunt. A rifle may not be legal for turkey, and is not legal for ducks.

For versatility, the Rem 870 and Mossberg 500 shotguns that you mentioned are hard to beat. Both are reliable and are available with easily changeable screw-in chokes.
Auto loading shotguns absorb some of the recoil (to work the action), but they can be finicky with the type of ammunition shot through them.

I am not a fan of buckshot for hunting deer as the probability of wounding increases with the range of the shot. Slugs have and can be shot through smooth shotgun barrels, however accuracy is generally better if shot through rifled barrels. Both Remington and Mossberg make easily interchangeable rifled barrels for the shotguns that you mentioned.

All shot pellets (of the same size and muzzle velocity) travel the same distance regardless of the gauge shotgun they are shot from. The difference is pattern density which is controlled by the choke (restriction at the muzzle) and the total number of shot pellets in the shell. Obviously the 12 gauge, being larger diameter, will hold more pellets than the smaller gauges. Usually you will also find a larger variety of 12 gauge shells than you will find in the smaller gauges.

Recoil is affected by weight of the projectile (shot or slug), powder charge, and the weight of the gun. (Recoil can also be modified by such things as porting or muzzle brakes, installing in-stock recoil reducers, back boring, and lengthening the forcing cones, but these things are not generally done for first time shotgunners or casual once a year shooters.)

A common fallacy is for a beginner to get a 20 gauge shotgun because it is smaller and weighs less than a 12 gauge gun. Then the shooter will shoot high velocity or magnum loads and be pounded by the recoil of the lightweight gun.