I've taken elk and deer with the 308 with no problems but given a choice I like a larger cartridge. But, if it's going to your only rifle for a while I'd recommend the the 308 or the 7mm-08 over a magnum just because you'll shoot it more. Get a bolt action rifle over an autoloader or a lever with a good variable power scope in the 2-8X or 3-9X range.

I've been doing this for a long time and here's what some left handed rifles look like.

Win M70 375 H&H, Ruger 7mm Rem mag, Savage 300 WSM, Remington 308, Savage 22.250, Savage 17 HMR, Stag 223, CZ 22 lr, Montana Rifle Co 6.5x284.

Here's the same 6.5x284 on the bottom with a couple of left handed Savages in 22.250 and 243 set up for long range shooting.