I also agree that the .30-06 is a good cartridge for deer and elk, and for most other North American game animals and most African plains game animals.
I started hunting elk when I was going to college and lived in Northwestern Colorado. Back then, resident deer and elk tags were over the counter, and for 10 years, I easily killed my mule deer and elk every year with my Herters .30-06.
I've had 3 Rem 700 rifles. All are accurate, and have been on African hunts at least once. One was an ADL, the other 2 are BDLs. I prefer the BDL because of the ease of unloading with the floor plate, although I don't like Remington's cheaply made aluminum floor plate. I also like the 2 position side safety.
Gun fit is also important, but stocks can be modified to better fit the shooter.
Recoil can also be reduced with proper gun fit, a better recoil pad, an in-stock recoil reducer, and/or a muzzle brake. Some cartridges, like the .30-06, are also available in reduced recoil loads, and lighter bullet weights will recoil less. By handloading you can also make reduced recoil loads.
Howa and Weatherby Vanguard rifles are also good and are inexpensive.