Warrior start with different weapon skills depending on their race, though most can train to the other skills if they would rather use those.
Maces, Axes and Swords are the most common weapons for a warrior. If you fight with a two-handed weapon, you will do more damage but will not be able to carry a shield, so think about your play style again when deciding which weapon type is best for you. If you want to play with a weapon you are not currently skilled with, the following cities have trainers that can sell you the skill you need.
Alliance
Darnassus – Daggers, Staves, Bows, Fist Weapons, Thrown Weapons
Stormwind – Swords, Crossbows, Polearms, Staves, Daggers

Ironforge – Guns, Axes, Crossbows, Daggers, Fist Weapons, Maces, Swords, Thrown Weapons
The Exodar - Crossbows, Daggers, Swords, Maces
Horde
Orgrimmar – Axes, Bows, Fist Weapons, Daggers, Spears, Staves
Thunder Bluff – Maces, Guns, Staves, Daggers
Undercity – Swords, Polearms, Crossbows, Daggers
Eversong Woods - Daggers, Swords (1H & 2H), Bows, Polearms
Silvermoon - Daggers, Swords (1H & 2H), Bows, Polearms
Keep in mind that there are a few weapons not available to you as a warrior, but these are fairly obvious, such as magic wands. When you reach the city that has the weapon training you need, ask any guard where the weapon trainer is and he will direct you straight to them. It costs 10 silver to train in any weapon you are not familiar with, regardless of which city you are in.
Because the range of weapons is so varied for a warrior, there is a lot of discussion on which type is the best to use. I personally enjoy polearms, which do a lot of damage fairly quickly, and I recommend getting one ranged skill of your choice, just to have it for pulling purposes and other situations where running straight into melee may not be the best idea. Be aware, however, that if you pick a ranged weapon that requires ammunition, such as a bow or gun, you will need an ammo pouch which will eat up one of your bag slots. It is usually a good trade off to have in my experience.
Daggers are usually not the best for warriors because of their low damage, but they are very fast and if you have a warrior built for speed, it may be a better off-hand than another sword or axe. Staves are almost useless as they take up two hands but deliver minimal damage next to a two-handed axe, mace or sword. Fist weapons also do minimal damage, but they can be fun to play with and what warrior doesn’t like a good brawl? Experiment and figure out which weapons you enjoy the most, then take a look at the list of where to find the best weapons of that type at the back of this guide.