Ya know, for a first timer, you might want to consider a double barrel, either a side-by-side or a built-wrong (over-and-under

.
They can be got pretty cheap, and are reliable as dirt, and often lighter to carry for upland bird hunting.
And yes, a good twelve gauge shotgun is good for everything you could care to hunt, from grouse to deer and back again. If you want to make it a truly all-purpose gun, though, better off with the pump gun, and with the 870, you can buy special-purpose barrels for it later, for slugs, for example, if you wanted to use it for deer hunting, or a tight one for turkey.
To get into the terminology, '12' is the gauge, the diameter of the ammunition, not the 'load', the composition of said ammo. The load is composed of two things: a size/diameter of pellet and the number of them, measured as a total weight of shot. For pheasants, if you can still use lead shot for upland game where you are, something in the 4-6 range of shot size in a 2 3/4' shell should provide you with the fixings for 'pheasant under glass' without too much fuss.
Myself, I use a 20 gauge when ever I have the rare hankering to pay-shoot pheasant (not native here in my neck of the woods), and use a 3' mag #6, as that is often the largest shot size allowed, so I try to compensate by using a heavier shell (the 3' mag).
Grouse here fold up nicely under 2 3/4' #7s (Federal general purpose shells from WalMart

in the 20.
A sixteen gauge is an odd bird; very popular, but hard to find 'fodder' for. I'd go bigger or smaller, if I was you; a 12 or a 20. And if you want it to be a one-gun arsenal, go for a pump 12.