There are lots of trainers located all over the world that teach you many things. However, they can be a little hard to find at times and some of the trainers are actually hidden in obscure places. Here's a list of the various trainers ranging from Mounts to Weapons to Professions.
Weapon Trainers
Characters start out with various weapon skills already learnt at level 1, but that is hardly the full range of weapons your character is capable of wielding. To use weapons that you currently cannot use, you will have to visit a weapon trainer teaching the appropriate weapon skill and pay a small fee.

Weapon trainers can only be found in cities, once you have reached a city, finding the weapon trainer is just a matter of asking a city guard where he/she is located.

Once you found him he can either train you or advise you on where other trainers are. If you want him to train you he'll show you a list of which weapon types he can teach.

City Faction Weapon Types
Darnassus Alliance Daggers, Staves, Bows
Stormwind Alliance Swords (1h & 2h), Crossbows
Ironforge Alliance Guns
Sayoc, Orc Weapon Master, Valley of Honor, Orgrimmar
Horde Bows, Daggers, Fist weapons, one & two handed Axes and Thrown Weapons

Hanashi, Troll Weapon Master, Valley of Honor, Orgrimmar Horde Bows, one and two handed Axes, Staves and Thrown Weapons
Thunder Bluff Horde Maces (1h & 2h), Guns, Staves, Daggers
Archibald, Undead Weapon Master, War Quarter, Undercity
Horde Crlossbows, Daggers, one and two handed Swords and Polearms

Mount Trainers and Vendors
When your character reaches level 40, he/she will be eligible to buy a mount. While mounted, you will travel 60% faster than normal. Each race has its own unique racial mount, for example, Trolls ride Raptors while the Tauren ride Kodo beasts. Before you can ride a mount though, you have to learn the riding skill specific to the mount first, such as Raptor Riding or Kodo Riding. The cost of learning how to ride is fixed at 20g, and the mount itself costs 80g.

Players can always learn how to ride mounts of their own race, to learn how to ride mounts of an allied race in your faction you have to reach the Exalted(?) reputation status with that race by doing quests and other things for that race, otherwise the mount riding trainer will refuse to teach you. Additionally, players cannot learn how to ride mounts of the opposite faction as the NPC trainers are hostile (and cannot be Mind Controlled). Due to the Tauren's huge size, they are restricted to only be able to ride Kodo Beasts and cannot ride other mounts. Any player can ride a Kodo though, not just a Tauren.

Mount vendors are located right next to the mount trainer itself. They sell mounts of various colours and also sells level 60 versions of mounts that travel 100% faster instead of 60%. However, these magnificent beasts cost 1000g to purchase. The Humans have 3 Horse Vendors while the others only have 1.

Race Faction Mount Trainer/Vendor Location
Human Alliance Horse Eastvale Logging Camp, Elwynn Forest
Vendor 2: Menethil Harbour, Wetlands
Vendor 3: Southshore, Hillsbrad Foothills
Vendor 4: Theramore, Dustwallow Marsh

Dwarf Alliance Ram Amberstill Ranch, Dun Morogh
Night Elf Alliance Night Sabre Darnassus, Teldrassil
Gnome Alliance Mechanostrider Steelgrill's Depot, Dun Morogh
Orc Horde Wolf Orgrimmar, Durotar
Troll Horde Raptor Sen'Jin Village, Durotar
Tauren Horde Kodo Beast Bloodhoof Village, Mulgore
Undead Horde Undead Steed Brill, Tirisfal Glades

There are other mounts for sale in the world that any race can get but these NPCs are hard to find, and also require the players to acquire a certain amount of reputation with them before they will even sell these mounts.

Profession and Secondary Skill Trainers
Professions and secondary skills are non-combat skills that players can learn in order to improve and further customize themselves. A player may choose to learn up to 2 professions and any number of secondary skills. For more information regarding these, visit our Professions section.

To learn CLvl Req Allows Skill Cost Learn it from
Apprentice 5 1-75 10c Journeyman Trainer
Journeyman 10 1-150 5s Expert Trainer
Expert 20 1-225 50s Artisan Trainer
Artisan 35 1-300 5g Master Trainer

Just because a trainer is called an <Expert Leatherworker> does NOT mean it teaches the Expert skill level. In fact, such a trainer is only able to teach the Journeyman skill level and Journeyman related Leatherworking recipes.

Apprentice skill levels can be learnt in most starting towns in the beginner zones as well as in the cities. Journeyman skill levels can also be found in cities and some towns all over the world. To find a trainer of any kind in a major city, simply ask a city guard for instructions and they will put a market on your map (and minimap)!

However, there is only one trainer that teaches the Expert skill level and one trainer that teaches the Artisan skill level in the world per faction. Players will have to find these two NPCs (Artisan and Master Trainers) in order to advance their skill beyond 150 and 225 respectively.