Weapon-Skill Trainers
Alliance
Swords + 2H Swords
Stormwind - Old Town (Garrison)
(take a left when you see the first rogue trainer)
Maces + 2H Maces
Ironforge - Mystic Ward (by Talent Master)
Axes + 2H Axes
Ironforge - Military Ward (by Warrior Trainers)
Daggers
Darnassus - Warriors Terrace (by blacksmith)
Ironforge - Forlorn cavern
Staves
Darnassus - Temple of Elune (just inside)
Stormwind - Mage quarter, by the staves shop
Spears
Darnassus - Warriors Terrace (by blacksmith)
Fist Weapons
Ironforge - Forlorn Cavern (in Rogue building)
Darnassus - Warriors Terrace (by blacksmith)
Polearms
Stormwind - Old Town (Garrison)
Crossbows
Stormwind - Old Town (Garrison)
Ironforge - Military Ward (by Warrior Trainers)
Bows
Darnassus - Cenarion Enclave (by Hunter Trainers)
Guns
Ironforge - Military Ward (by Warrior Trainers)
Thrown
Darnassus - Cenarion Enclave
Ironforge
Horde
Swords + 2H Swords:
UnderCity near warrior/priest trainers, across from them on the other side of the moat in the middle of the vendors area
Maces + 2H Maces:
Thunder Bluff - near warrior trainer
Axes + 2H Axes:
Sen'jin (troll) village
Ogrimmar - near warrior trainer
Daggers:
Orgrimmar - Cleft of Shadows (Rogue and Warlock area)
Thrown:
Orgrimmar - Cleft of Shadows (Rogue and Warlock area)
Staves:
Ogrimmar – near the Mage trainers in the Valley of Spirits
Thunderbluff - on the elder rise
Spears:
Ogrimmar - hunters lodge
Fist:
Orgrimmar - wanders the Valley of Honor (outside/near warrior trainer)
Polearms:
UnderCity - in front of warrior/priest trainers, right side at start of ramp
Crossbows
UnderCity - near rogue trainers
Bows
Ogrimmar - hunters lodge
Sen'jin (troll) village
Guns
Thunder Bluff - Hunter Rise - warrior trainer
Profession Trainers
Most major cities have a basic trainer in each profession. Ask your friendly city guard for directions

Trainers train you in their level of expertise, so you need to be at the skill level below the trainer to be able to learn from him. (if the trainer teaches the 150 level, you need to be at the 75 level). If you are just starting a profession, seek out a trainer in one of the major cities. When your skill level rises, the hunt for the right trainer gets more difficult. For a detailed description of the trainers and what skill level you need to learn from them, see the bottom of this post.
To find the trainers in each town, ask the guards for directions:) They will put a flag on your mini-map and world map (props blizz:).
Alliance
Alchemy
Journeymen: Ironforge (tinkertown), Stormwind (Mage quarter)
Expert: Ironforge (tinkertown)
Artisan: Darnassus
Master: Feralas
Blacksmithing
Journeyman: Ironforge, Goldshire, Stormwind (dwarf quarter)
Expert: Stormwind (dwarf quarter), Duskwood
Artisan: None confirmed yet
Weaponsmith Master: Stormwind (dwarf quarter)
Armourer Master: Booty Bay
Weaponsmith Questgiver: Ironforge
Armourer Questgiver: Ironforge
Enchanting
Journeyman: Ironforge, Stormwind (close to mage quarter)
Expert: Feralas, Stormwind
Artisan: Elwynn Forest (Tower of Azora)
Master: Uldaman (within instance, near bats)
Engineering
Journeyman: Stormwind (dwarfen district), Ironforge (tinkertown), Loch Modan (on the bridge)
Expert: Stormwind (dwarfen district), Ironforge (tinkertown), Darkshire
Gadgetzan (Tanaris)
Artisan: Ironforge (tinkertown)
Master: None confirmed yet
Herbalism
Journeyman: Stormwind (Mage quarter), Menethil Harbour
Expert: Stormwind (Mage quarter), Menethil Harbour
Artisan: Booty Bay, Lakeshire, Stomrwind (Mage quarter), Menethil Harbour
Master: Booty Bay
Leather
Journeyman: Darnassus
Expert: Stormwind (old quarter), Darnassus
Artisan: Darnassus
Master: Aerie Peak (Hinterlands)
DragonScale: Azshara (near path to the SE beaches)
Elemental: Felwood, on the north side of the vale with the earth and water elementals (not confirmed)
Mining
Journeyman: Thelsamar, Duskwood
Expert: Thelsamar, Duskwood
Artisan: Thelsamar, Duskwood
Skinning
Journeyman: None confirmed yet
Expert: None confirmed yet
Artisan:Darnassus, Redrige Mountain, Stormwind (old quarter)
Master: None confirmed yet
Smithing
Journeyman: None confirmed yet
Expert: None confirmed yet
Artisan: None confirmed yet
Master: Booty Bay
Tailoring
Journeyman: Ironforge, Elwyn Forest (Eastvale Logging Camp), Daranassus
Expert: Ironforge, Stormwind (Larsons Clothier, Mage district)
Artisan:Stormwind, Daranassus
Master: Theramore Island
First aid
Stormwind - Cathedral of Light, Right wing
Ironforge - Near great forge, same building as herbalist
Menethil - 2nd floor at the inn
Cooking
Goldshire - In the Inn
Ironforge - Copper Kettle
Stormwind - Pig and Whistle, Old Town
Darnassus - (dont know where yet, probably the inn)
Lakeshire - By Chef Brianna (West end of town)
Fishing
Thelsamar - Just north-east of town, by the water
Horde
Alchemy
Journeymen: Ogrimmar (The Drag)
Expert: Undercity (Apothecarium), Booty Bay
Artisan: Undercity (apothecary)
Master: Swamp of Sorrows (Stonard)
Blacksmithing
Journeyman: None confirmed yet
Expert: None confirmed yet
Artisan: None confirmed yet
Master: Booty Bay
Enchanting
Journeyman: Undercity (apothecary)
Expert: Undercity (apothecary)
Artisan: NEW Sunrock Retreat (without the flight path)
Master: Uldaman (within instance, near bats)
Engineering
Journeyman: None confirmed yet
Expert: None confirmed yet
Artisan: None confirmed yet
Master: Gadgetzan (Tanaris)
Herbalism
Journeyman: Feralas (Camp Mojache), Grom'Gol
Expert: Feralas (Camp Mojache),Grom'Gol
Artisan: Booty Bay, Orgrimmar, Feralas (Camp Mojache), Grom'Gol
Master: Booty Bay
Leather
Journeyman: None confirmed yet
Expert: Grom'Gol
Artisan: Thunder Bluff, Feralas (Camp Mojache)
Master: Feralas (Camp Mojache)
Tribal: Stranglethorn Vale (across the river from Grom'Gol)
DragonScale: Badlands (eastern portion)
Elemental: Hinterlands near Fire Ele's (not confirmed)
Mining
Journeyman: None confirmed yet
Expert: None confirmed yet
Artisan: None confirmed yet
Master: Gadgetzan (tanaris)
Skinning
Journeyman: Feralas (Camp Mojache)
Expert: Feralas (Camp Mojache)
Artisan: Feralas (Camp Mojache)
Master: None confirmed yet
Smithing
Journeyman: None confirmed yet
Expert: None confirmed yet
Artisan: None confirmed yet
Master: Booty Bay
Tailoring
Journeyman: Undercity (mage quarter)
Expert: Undercity (mage quarter), Booty Bay
Artisan: Undercity (mage quarter)
Master: Tarren Mill (Hillsbrad)
First Aid
Brill
Undercity
Orgrimmar - valley of spirits
Cooking
Grom'Gol (trains up to artisan
Class Trainers
currently just a list over all the major cities for the factions. Some of these trainers might not exist in all cities.
Druid
Stormwind
Undercity
Thunderbluff
Darnassus
Feralas (Camp Mojache)
Hunter
Stormwind
Ironforge
Orgrimmar
Thunderbluff
Darnassus
Grom'Gol
Mage
Stormwind
Ironforge
Orgrimmar
Undercity
Thunderbluff
Paladin
Stormwind
Ironforge
Theramore
Priest
Stormwind
Ironforge
Orgrimmar
Undercity
Thunderbluff
Darnassus
Rogue
Stormwind
Ironforge
Orgrimmar
Undercity
Thunderbluff
Darnassus
Booty Bay
Shaman
Orgrimmar
Thunderbluff
Undercity
Warlock
Stormwind
Ironforge
Orgrimmar
Undercity
Thunderbluff
Stonard
Warrior
Stormwind
Ironforge
Orgrimmar
Thunderbluff
Undercity
Darnassus
Stonard
Talent removers
note that these will NOT be here for retail, only for beta testing purposes.
Ironforge - The mystic ward
Stormwind - Cathedral
Undercity - Royal Quarter
Orgrimmar - Valley of wisdom, next to thrall
Darnassus - Inside the temple
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Detailed description of Profession trainers and what they teach (thanks to william):
Tradeskill trainers train just the level below their level. If the trainer is more than one level above you he will not train you in anything. A Journeyman Trainer will teach you the initial Apprentice tradeskill and the skill 1-75 items an Apprentice can make, so long as you are at least level 5. The higher level trainers won't teach you anything. You can go to the Journeyman Blacksmith in the Dwarf District in Stormwind to learn Apprentice Blacksmith, but the Expert Blacksmith next to him won't talk to you until you have at least skill 50 in Blacksmith.
An Expert Trainer will train an Apprentice with skill at least 50 and at least level 10 up to a Journeyman and train the skill 76-150 items a Journeyman can make. An Artisan Trainer will train a Journeyman with skill at least 125 and at least level 20 up to Expert and train the skill 151-225 items an Expert can make. A Master Trainer will train an Expert with skill at least 200 and level at least 35 up to Artisan and train the skill 226-300 items an Artisan can make that do not require a specialty. Master Specialist Trainers teach their specialty, usually after the completion of a quest, and their specialty items to Artisans.
Note that most of the basic trainers are not listed, they can normally be found in the major cities.
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What type of weapon skills can my class train?
Druids
Staves, Mace 1H and 2H (with training?), Daggers (with training), Spears (with training), Unarmed Weapons (with training)
Hunters
Axes, Daggers, Guns or Bows, Crossbows (with training), Fist Weapons (with training), Spears (with training), Staves (with training), Swords (with training), Thrown (with training), Two-Handed Axes (with training), Two-Handed Swords (with training)
Mage
Staves, Wands, Daggers (with training), Swords (with training)
Paladins
One- and Two-handed Maces, One- and Two-handed Swords (with training),One - and Two-handed Axes
Priests
One-Handed Maces, Daggers, Staves, Wands
Rogues
Daggers, Thrown, Swords - 1 hand (with training), Maces - 1 hand (with training), Fist Weapons (with training) Bows (with training), Crossbows (with training), Guns (with training)
Shamans
One-Handed Maces, Staves, One-Handed Axes (with training), Daggers (with training)
Warlocks
Daggers, Wands, Staves (with training), Swords (with training)
Warriors
Axes (2H with training), Maces (2H with training), Swords (2H with training), Daggers, Spears (with training), Polearms (with training), Bows (with training), Crossbows (with training), Guns (with training), Staves (with training), Thrown (with training), Fist Weapons (with training).