Good choices you have and good luck with your new addiction. I personally am a 270 caliber person. I like all those but have shot the 270 to 300 yds with virtually no adjustment in aim. I have for severely years now shot a Remington 700 series short mag 270 that is a tack driver out of the box. But for a new hunter I would have to suggest one of my favs, the 243. I've shot more deer and most have dropped on the spot with that caliber than any other. Easty recoil and good out to 200 yds with ease. Don't matter, you will buy more over the years if you are like most of us. Have fun. BTW, my first deer rifle in my poor, hand me down days was a British 303 that weighed a hundred pounds and a full barrel stock. Had a flip up, fix sites for shooting out to 1000+ yds!!!! Not sure how many men it had killed in the War, before I inherited it from my Grandfather.
I got one of those Enfield british 303 guns as well. Can't use it to hunt anything here, but certainly an interesting gun. Mine came back from WWII as well and was sporterized for hunting by my grandfather. Sure is a heavy thing.....I guess if you ran out of ammo you had a club!
As for the OP - I was in the same boat and I think l if I get me a new gun for ME - it will be a .308 and you can find good package guns from Ruger, Savage and Remington for a real decent price as well. Lots of ammo options and should handle pretty much anything you want to hunt in the lower 48 from a big game perspective.