Critical has a better chance of headshots while piercing can penetrate armored enemies easier and also allow the bullet to strike through multiple targets should they be lined up. For a handgun you'd probably want to go for the critical, at least if you're just starting out. The less ammo you need to expend on enemies the easier it'll be for you, although there is a horde of handgun ammo throughout the entire game.
+Gives you a nice quick kill which is especially useful in The Mercenaries/Versus Slayers - A higher chance of a parasite showing up - You need luck to get a Critical
+ Can penetrate armor and shoot through enemies - Unless you're being swarmed by Majini, I doubt you will be shooting through 2 enemies at once very often - Armored/shielded enemies don't show up until Chapter 3
There isn't a higher chance of plagas turning up with critical. Piercing on the handgun you just shouldn't worry about. Don't consider the rarer occasion when majini line up and when there's armor, use switch to rifle.
Get good with handgun headshots and you'll save a lot of ammo for other weapons for when you need it.
I SHOULD be? The lighting in the game makes it a bit difficult to see anything from a distance and the laser sight on the guns is no exception. I can do it with a rifle easily but a handgun? It takes a bit of timing to do that if I don't want to miss a shot each time. The enemies tend to group together anyways so piercing is more beneficial.