If you've not shot a lot, or unless you plan on doing a lot of shooting before you go hunting, magnum cartridges are probably not for you.
With that out of the way, just about anything in the 30 cal or 284 caliber family will do nicely, and without any punishing recoil (for new folks, this is a thing).
so, in no particular order...
30-06
.308 Winchester
7mm-08
280 Remington
7x57IS mauser.
All of them will get you comfortably out to 300 yards on most game, but make sure you are using quality ammo - Barnes and Nosler are great. I like Hornady myself, but quality ammo is very easy to obtain for all of them, and the price won't break your wallet, either.
As
@Art Lambart II says, the 35 Whelen is also an excellent choice. There are several other 35 caliber cartridges, but I would avoid them unless you want one for maybe your 4th or 5th rifle into a collection of rifles.
Also, you don't need to spend a fortune on the rifle, either. There are lots of fine-shooting rifles chambered in those cartridges that cost less than 1000 USD, even less than 700 USD. Put a good scope on your rifle - quality scopes can also be had for $500.
Above all, practice before you go. A lot. 20 rounds is not a lot of practice. At the end of the day, a rifle is simply a tool. YOU are the weapon. You could spend your life's savings on a really high end rifle, and top it off with Zeiss or Swarovski optics, but if you don't practice, you'll miss your target, or worse, wound it and have it run off and die in misery.