Here is the ultimate guide to the monsters in Diablo. Follow closely as this will save you from death in the depths of the dungeons. Be sure to also look at the monster table at the bottom of this page.

When all else fails, find a good, protected vantage point where you can't be surrounded by the enemy and do what comes naturally--spells for Sorcerers, arrows for Rogues, and a giant sword, club, or axe for Warriors. Make sure you have a Town Portal spell on a hot key--nothing can kill you if you have a quick getaway plan.

Take note: just because a monster's description says it can resist fire, that doesn't mean fire can't hurt it--especially Fire Wall spells. It will still take damage, just not 100 percent of the damage. Immunity on the other hand, means the creature will not be affected at all.
For persistent skeleton problems, use Holy Bolt. It kills skeletons dead. Rogues without Holy Bolt should do what they always do--start plucking on their bows. Warriors without Holy Bolt should use a club, mace, or maul. They're much more effective against animated bones than are sharp-edged weapons.

When you first run into a mass of Steel Lords, Black Knights, and Blood Knights, you're going to get your tail kicked. If you're a Warrior, cast a Guardian spell and engage these in hand-to-hand in a doorway or narrow entrance. That way they can only get to you one at a time, and your Guardian (or two) will keep picking them off the whole time. Rogues should keep on moving away from these hefty buckets of metal. They're slow enough that you can pick them off as you run away. Sorcerers, you'd better break out your spell arsenal. Fire works well against the Black Knights and Blood Knights, and Chain Lightning works well against Black Knights and Steel Lords.

Skeletons with bows (also known as the Burning Dead) are dangerous indeed. Again, use Holy Bolt or ranged weapons if you can. If you're a Warrior, make sure you make taking out these Burning Dead archers a priority. Otherwise you'll take a ton of unnecessary damage from their unholy arrows. Although they are fire-resistant, Fire Wall can still hurt these guys.
You'll run into Red Storms that shoot lightning bolts in the lower depths of the Catacombs. Keep moving around when you face these fiends, and make sure you have your resistance to lightning at full-tilt.

Horned Demons do massive amounts of damage with each charge. If you have ranged weapons, try to keep as much distance as possible between you and them. Spells that work nicely against them include Chain Lightning and Charged Bolt. If you can't keep them away, work in close against them; if they can't build up speed, they can't charge you.

With their ability to disappear and reappear at will, Illusion Weavers are extremely difficult to defeat. When fighting them hand-to-hand, characters should left-click-attack once and then move away. Once they reappear, left-click-attack them again.