I'll assume a person is pursuing the common American game animals here.
.22 long rifle, marksmanship, plinking, and small game, does it all. Rifle and handgun please.
.223 for varmint, predators, target work and even home defense look no further. Ammo is relatively inexpensive, and plentiful.
.243 replacement for the .223 if you are not a varmint shooter. Will work on deer as well as on coyotes.
7mm-08 As fine a whitetail cartridge as could be invented. Will travel out west for pronghorns, sheep, and muleys as well.
A .30 calibre... Might as well finish pee'ing off everybody. .300 Winchester Magnum, if you can see it you can kill it
. For not that much more recoil you have a 30-06 beat every way to sunday. Deer, Antelope, Bears, Sheep, Elk, Moose, it's the Honey Badger of hunting rounds... it don't give a ... care kills 'em all.
9mm cheap reliable fodder for both home defense and plinking.
.357 magnum, mo' better than a 9mm and comes in some of the finest handguns ever made. Feel free to sell the 9mm to finance a .357 mag.
.338 Winchester Magnum, 'cause sometimes a .300 WM don't kill stuff dead enough. That's all I got to say about that.
45-70 'cause this is still America and if you are an American you need a lever action rifle and what could be better than a levergun fed with a good ol' fashion black powder cartridge modernized to run on smokeless?
.50 sidelock blackpowder rifle. Easier to understand where you are when you know where you came from.
For those on a budget I can pare this down a bunch.
East of the Mississippi .22 LR and a 30-30, you southern boys will have to learn to walk and stalk.
West of the Mississippi .338 WM and a .44 mag. handgun. Elmer Keith chose those two and I sure ain't gonna go against his judgement on the subject.