About the only thing I’ve learned in terminal ballistics as the volume of game shot increases, by my clients and their guides or in my own adventures, is speed kills as long as half or more of the bullet holds together. Muzzle velocity is not the first place to look, impact velocity is, I like to see it at 2200-2400+ at the animal. That can handily be a .308 at all reasonable hunting ranges, so we’re not talking all magnums and 28 Noslers.
If I had to pick a cartridge with the most outsized performance on game relative to recoil, it’d be several but they all have speed in common; .270 Winchester, 7mm Mag, .270 WSM. .300s, .338s, the bigger 9.3s, and .375s do it too at the cost of greater recoil.
My suggestion is to get a normal rifle you can shoot well in a hardware store shelf chambering, you couldn’t go wrong with a .308, .270, 7 Mag, or .30-06.