Good deer cartridges for the recoil sensitive:
.243 Win.
6mm Rem.
.257 Roberts
.25-06 Rem.
.260 Rem.
6.5x55mm Swede
7mm-08 Rem.
7x57mm Mauser
Of these, the most commonly available (rifles and ammunition) are the .243 Win, .25-06 Rem, 7mm-08 and 7x57.
If ranges will likely be shorter, the .243 is the most gentle, and still a perfectly capable deer round at 300 yards.
If expecting longer shots (or just wanting a flatter trajectory) the .25-06 is pretty much the undisputed champion of deer rifle chamberings.
If elk are ever to be hunted, I'd suggest the 7mm-08 or 7x57. .25-06 is fine for elk with proper bullet selection and good shot placement, but the heavier 7mm bullets give a better margin of error.
Some have suggested the .270, which is certainly a good round. But the .270 Win (and .280 Rem) will kick harder than a .308 in a given rifle.