Pistols.

Pistols:
Well, it depends.  Is it magazine-fed or does it have a revolving cylinder?  What will it fire?  How reliable is it?  Is its ammunition common where you are now and where you are going?  Is it accurate?  Can you handle the recoil? Some of these questions overlap a bit, but you still need to ask them, and more.
  What type of pistol are you thinking of?  Each has interesting advantages over the other: revolvers are reliable, have few moving parts, are easy to fix (compared to pistols), can chamber large rounds easily, and have a completely customizable grip, while pistols have high capacity, are almost as reliable as revolvers (some pistols more so than others), and are now used by more people.  Revolvers can be reloaded by hand much more easily than pistols, but a pistol can simply change magazines if one is available pre-filled.  If you want my opinion, I'm inclined to say that a pistol would be a better idea.  Pistols have a clear lead in time spent reloading versus ammunition replenished, so long as you have spare magazines.  They are also, as I said, becoming as reliable as revolvers.  And the spare parts issue isn't really that dire; if you take care of your pistol, you will be able to use it many thousands of times--many more than you'd expect to fire it in a zombie outbreak--without incident.  Still, that's just my preference.  Try both types and decide which you like more.  Now, on to cartridges!
  There are two generally accepted ways to measure the size of a cartridge: caliber, which measures by hundredths of an inch, and metric measurement, which uses millimeters.  Between the two, metric is more informative.  Every metric round tells you its width and length, in that order, separated by an x.  For example, a 9x19mm cartridge has a diameter of 9 millimeters and a length of 19 millimeters.  This helps you figure out which cartridges you can fire and which ones you can't. Even then, though, you have to be careful: most cartridges have nonstandard "loads," with either more or less powder.  For example, an overpressure cartridge--a cartridge filled with extra propellant--can damage a gun not designed to fire it because of the additional pressure in the barrel (hence its name), while a standard cartridge may not fully push back the slide of a gun designed for overpressure loads.  Fortunately, most overpressure ammo is marked with a little "+P" sign somewhere on its base, and really high-pressure stuff is marked with a "+P+." That stuff can blow a normal barrel apart!  And that's why you should always know what your gun takes for ammunition.
  Finally (for now), reliability.  Read online reviews on the pistol you're planning on buying.  Go to gun shops and ask about its reliability.  Ask owners about how it feels, how often they shoot it, and how many times it's failed versus how many shots it's fired.  It doesn't take all that much effort, but it's the main difference between choosing a reliable piece and a comparative junker.  And reliability isn't the only thing you need to consider: think of the features you want, the recoil, the report, and how many are in circulation.  Colt 1911s, Beretta 92Fs and 92FSs, and Glocks, have all proven themselves to be reliable, durable, and accurate, and have many "relatives" that you can salvage for parts (especially the Glocks).  Take the time you need to research the pistols you've got in mind, because if you pick the wrong one, you won't get a retry.
  Calibers and cartridges:
There are a lot of different cartridges from which to choose.  All have their unique advantages, circumstantial perks that you can use with certain tactics.  However, I find that these six cartridges in particular are suited to the zombie apocalypse.  Maybe you'll like one.
  First, the .22 Long Rifle (abbreviated to .22 LR).  It's a small cartridge, but it's racked up as many brownie points as any other cartridge.   First, it's easily the single most common cartridge available.  Pretty much every gun store has tons of boxes in the ammo section.  Second, it's tiny; you can carry ridiculous amounts of .22 cal with ease.  You won't run out and you won't be loaded down by the ammo.  It's true that this cartridge doesn't have as much punch as larger calibers, but, as you've probably heard from numerous sources (I certainly have), it'll rebound in your enemy's skull, causing massive damage to his or her brain.  Even if it passes through the enemy's head without that brain-scrambling rebound, you'll still be shooting him in the head.  And although the .22 might not penetrate a skull at long range, we're talking about using this in a pistol, a close-range weapon.  The .22's lack punch won't be an issue at fifty feet, or even fifty yards.  Ultimately, despite its deficiencies, the .22 LR is an excellent cartridge.
  Next, the 9x19mm.  This is, I'll openly admit, my favorite cartridge.  First, while it does plenty of damage to the average infected head, it's small enough that you can cram around 16 or 17 rounds of it into a pistol without extending the magazine.  It's light, so you can keep plenty with you, and it'll protect you from both raiders and infected.  Finally, the 9x19mm cartridge has been around since before World War II, on both sides of the conflict.  It did its job then, and still does it now, maybe even better if you consider all the advancements we've made in gun technology.  Plus, it's metric.  Now, on to the next round!
  The .40 Smith and Wesson cartridge (abbreviated .40 S&W) is also a solid chambering.  it's a lot like the 9x19, but just a liiitle bit larger at about 10 millimeters wide.  This, in turn, means that it takes more flesh and bone with it as it passes through a zombie's body, creating a larger wound channel and destroying more flesh.  And while it travels more slowly than a 9x19, it still gets to--and through--a body very, very quickly.  Whatever your preference, you can't say that the .40 S&W isn't a solid round.
  Next, we've got the .45 ACP.  This is quite possibly the most recognizable cartridge in the US.  It served in the US army for close to a hundred years and in several commendable weapons, including the ever-popular Colt M1911 pistol.  In fact, the main reason for was because the US wanted to be able to share ammunition with its allies during a war.  Every country had to be able to able to make the same ammunition, and since the rest of the world is metric, we had to pick a metric round.  Anyways, the .45 ACP is a powerful, popular, reliable cartridge.  You can fit around 13 rounds into a double-column magazine without extending the mag past the butt of your pistol, and while it is heavier than other rounds, the .45 is still small enough that you can easily carry a few hundred without too much trouble.  Overall, this is an excellent round to choose.

  Finally the .357 Magnum and the .38 special round out this list of cartridges.  While they sound like they've got dangerously different specs, the .38 and the .357 actually have exactly the same case diameter.  When S&W made the .38 spl, they measured the case instead of the base of the exposed bullet.  The case had a .379 inch diameter, rounded off to .38 inches (hey, "thirty-eight" is easier to say than "three-seventy-nine").  The .357 Magnum, on the other hand, was measured at the bullet's exposed base, giving it a different measurement.  The upswing of all this?  The similarities give a revolver chambered for the .357 Magnum the ability to fire the .38 special.  However, this doesn't work in reverse: thanks to the .357 magnum's longer case (and the extra powder inside),  it's fired at a much higher pressure than its parent cartridge, and a revolver built for the .38 special can be damaged by that extra pressure.

  You probably noticed that I was talking about using the .38 spl and .357 Mag in revolvers.  This is because a magazine-fed pistol is designed for a specific diameter and length of cartridge, whereas a revolver can almost always shoot shorter cartridges, so long as they are of the same diameter as the cartridge for which they were chambered.  If you want to fire either one at your option, get a revolver (or a second pistol, but that's more weight).  Both cartridges are common enough that you'd probably have enough ammo for them, but if you want a revolver, I'd suggest the "Caliber .357, Magnum load."