Any decent adventurer needs weapons proficiency to survive their travels. Like a muscle, the more you use a weapon, the better you become at using it. In WoW Classic, that's represented by the Weapon Skills System. You'll learn all about it, getting better at WoW Classic gold farming, questing, and fighting along the way.
The Weapon Skills System
The Weapon Skills System keeps track of how much you've used a weapon. The higher levels mean you deal more damage and hit more often with that weapon type. Characters start out with a cap of 5 and power leveling up increases that cap by 5. By the time you reach level 60, you'll have a weapon skill cap of 300. That cap can be further increased through items, racial abilities, or certain skills.
To elaborate on its effects, having a higher weapon skill lets you damage enemies consistently. If you have a lower weapon skill than your target's defense, you'll end up dealing Glancing Blows more often. They don't deal as much damage and cannot be a critical hit. Naturally, it would take longer for you to defeat them that way.
For NPC enemies, if they have a 20 point difference between their weapon skill and your defense, they can deal a Crushing Blow. That deals 150% of the normal damage and can be a critical hit. What's more, if they hit you from behind, there's a chance you'll be Dazed and not be able to do anything.
For PvP, it doesn't quite matter in Battlegrounds. WoW Classic Weapon skills are maxed out, with a cap of your level times 5. Your enemies have their base defense calculated the same way, before adding gear effects. It does matter, however, in world PvP and duels.
How to Train Weapon Skills
Use the weapon. Every hit you make has a chance to increase the skill. With lower levels, it's near 100%. The chances will decrease to about 50% the nearer you are to the level cap. Having a weapon with a fast swing speed should make power leveling up your Classic WoW weapon skill faster. Intellect has a small bonus to leveling this, as well. You can use a weak weapon to attack a monster that would deal little damage. Defeating it isn't really needed, you just need to hit it enough times to level up the skill.
You may also talk to trainer NPCs in towns to train the skills. Each one has a different set of weapons they're willing to train you in. You can even ask them about the NPC who can teach you about a weapon they don't offer. Learning from a trainer will cost 10 silver pieces, except for Polearms, which cost 1 gold piece. Alliance trainers are in Darnassus, Ironforge, and Stormwind City. Horde trainers are in Orgrimmar, Thunder Bluff, and the Undercity's War Quarter.
Weapons have class restrictions, such as warlocks being unable to use bows. Similarly, only Mages, Priests, and Warlocks can use wands. Take note of what your class can use! Races also have proficiencies. It was mentioned above that some racial abilities raise the level cap of certain weapon skills. Make sure you make the most of it and pick a race and class that take advantage of these mechanics.
You can train all available skills at any level except for Polearms (when applicable). Fist Weapons have a similar limitation, though you must level the Unarmed skill instead. That is also the skill the weapons will utilize in calculations. They won't unlock a new skill.
And that's all for WoW Classic Weapon Skills! Get swinging for more hits and more damage. Efficiency is the key to quick leveling up or farming, so you need to level this. Good luck and keep enjoying World of Warcraft Classic!