.22LR - practice, plinking, small game, teaching new shooters, cheap!
.223 Rem - "real" round for 1/3 the price and recoil of common bigger centerfires. Great for recreational rifle shooting of all sorts -- any purpose that doesn't need a big bullet or 1000 ft-lb at distance.
.30-06 Spgfld - hunting most medium to big game. Readily available, bewildering variety of loadings in boxes and in books, great balance between recoil, price and capabilities. Every type of commercial rifle and action are mass-produced in this chambering, or in the shorter version, the .308, if you want a smaller rifle with slightly less flexibility in bullet weight.
12 Gauge - true, it's not technically a rifle round, but with a barrel or two on a decent shotgun, you have everything from a poor man's express rifle to a dove hunting gun. Newer rifled slug barrels with sabots turn in performance a lot like the old big-bore carbines.
.44 Magnum - also not technically a rifle round, but a respectable hunting round at iron sight ranges, and easily shared with a holstered revolver. Stand-in for a .30-30 out to 150 yards, and good for predator defense.
Honorable mention:
.50 muzzleloader - fun toy, hunting gun too.
I haven't messed with the real big centerfire stuff much, so I can't comment about that. .45-70 is attractive but I haven't shot it much. Ditto for .300 Weatherby -- the other day, I shot one with a muzzle brake, and it wasn't bad. Without the brake, forget it unless I have a specific hunting need.