Doom has all kinds of weapons to use.
Fist
The Fist is your weapon of last resort. It does minimal damage and has a slow refire rate, so you really shouldn't use it over any other weapon. However, its usefulness is increased when you've picked up a Berserk powerup, as this increases your fist damage to eight times normal.
Chainsaw
The Chainsaw is essentially an upgraded Fist, though it has a fair amount of major differences. The most obvious difference is that it has a very short refire rate - meaning that if you concentrate your attack on one monster, there's a very good chance that they will be unable to attack you because they're busy recoiling in pain. This doesn't mean that they won't even try, though - be especially careful around tougher monsters like Barons.
Pistol
Your basic starting weapon. The Pistol fires slowly, is rather weak, and isn't all that accurate. Firing the pistol slowly will ensure 100% accuracy, but this just makes the weapon that much less useful.
Shotgun
Your first truly good weapon, and possibly the most common weapon in Doom. The Shotgun fires a spread of 7 pellets, which individually do about as much damage as a single pistol bullet. You'll probably be using this weapon a lot.
Chaingun
A rapid-fire weapon that works wonders against lesser hellspawn, but becomes less useful against tough monsters. Individual bullets do about as much damage as the Pistol. Normally the Chaingun isn't very accurate, but there is a glitch that enables near-perfect accuracy. Instead of holding the fire button, simply tap it once to shoot off a 2-bullet burst. With good enough timing, you can keep tapping the fire button and be able to pick off foes from great distances using your new "chain sniper-rifle."
Rocket Launcher
A high-damage projectile weapon that is excellent for taking out large groups of difficult monsters (especially groups of Demons, Spectres, Cacodemons, or Lost Souls). Just keep your distance - don't want to blow yourself up! An interesting thing to note about the Rocket Launcher is that boss monsters (the Cyberdemon and the Spider Mastermind) are immune to splash damage, which means that the rocket launcher will do far less damage against them.
Plasma Gun
Here's a rapid-fire weapon that's far more satisfying to use than that dumb chaingun. Fires somewhat faster, but since it's a projectile weapon and not a hitscan like the chaingun, it'll be harder to aim over distances. I'd suggest saving the plasma gun for major crowd control operations - it eats through cells like nothing else. Well, nothing else except...
The BFG 9000
This thing cooks. After a short wind-up period, the Big F***ing Gun will shoot off a giant bolt of green plasma. The bolt itself does a lot of damage, but what most of the pain comes from is the "spray" damage. This is a little difficult to explain.
When the BFG ball explodes, it triggers a "spray" calculation that originates from your point of view. So, no matter where you're looking, when that ball explodes, any monsters within your point of view will take damage from the spray. Think of the spray like a large, invisible shotgun blast - monsters in front will be more likely to take spray damage, and closer monsters will be likely to take more sprays than others. Skillful use of the BFG can help to maximize damage against bosses and large groups of enemies.