All credit for this list goes to Guantlet Man 99, except for the few things that he left out.

Skill: Acrobatics
Location: Fighter's Quarters in the Waistworks in Vivec, Arena.
Person: Senyndie

Skill: Alchemy
Location: Propylon Chamber in ValenVaryon.
Person: Abelle Chiditte

Skill: Alteration
Location: Seryne Relas's House in Tel Branora.
Person: Seryne Relas

Skill: Armorer
Location: Bottom level of the Hawkmoth Legion Garrison in Ebonheart.
Person: Sirollus Saccus
NOTE: Due to a glitch, you can not talk about training w/ this character. The best you can do, is go to the trainer at Dren Plantaion.

Skill: Athletics
Location: Adibael's Yurt in the Kaushtababi Camp.
Person: Adibael Hainnabbibi

Skill: Axe
Location: Top level of Falensarano.
Person: Alfhedil Elf-Hewer

Skill: Block
Location: Courtyard of Buckmoth Legion Fort.
Person: Shardie

Skill: Blunt Weapon
Location: Vivec, St.Delyn Canton, Plaza, Abby of St.Delyn the Wise, Downstairs behind locked (lvl 30) door at the end of the hallway.
Person: Ernse Llerv

Skill: Conjuration
Location: Bottom level of the Temple in Ald'Ruhn.
Person: Methal Saran

Skill: Destruction
Location: Piernette's Farmhouse
Person: Leles Birian

Skill: Enchant
Location: Indoranyon
Person: Quorwynn

Skill: Hand-to-Hand
Location: Bottom level of the Holmayan Monastery.
Person: Taren Omathan

Skill: Heavy Armor
Location: Arena Fighters Training (Vivec).
Person: Seanwen

Skill: Illusion
Location: Dirty Muriel's Cornerclub (Sadrith Mora).
Person: Erer Darothril

Skill: Light Armor
Location: Andus Tradehouse (Maar Gan).
Person: Aerin

Skill: Long Blade
Location: Armiger's Stronghold (Malog Amur)
Person: Ulms Drathan

Skill: Marksman
Location: Missun Akin's Hut on top of Falasmaryon.
Person: Missun Akin

Skill: Medium Armor
Location: A boat in the Tel Fyr dock.
Person: Cinia Urtius
NOTE: Due to a glitch this trainer does not appear in the game. The best you can do is train it up to around 80, w/ the unarmored master trainer.

Skill: Mercantile
Location: Ababael Timsar-Dadisun's Yurt in the Zainab Camp.
Person: Ababael Timsar-Dadisun

Skill: Mysticism
Location: The Gateway Inn (Sadrith Mora).
Person: Ardarume

Skill: Restoration
Location: Voc Chapel.
Person: Yakin Bael

Skill: Security
Location: Hecerinde's House.
Person: Hecerinde

Skill: Short Blade
Location: Lucky Lockup Cornerclub (Balmora).
Person: Todwendy

Skill: Sneak
Location: Dreugh Jigger's Rest (Gnaar Mok).
Person: Wadarkhu

Skill: Spear
Location: Bottom floor of the Tower of Dusk (Ghostgate)
Person: Mertis Falandus

Skill: Speechcraft
Location: Mages Guild in Sadrith Mora.
Person: Skink-in-Tree's-Shade

Skill: Unarmored
Location: Vacant Tower (Dagon Fel)
Person: Khargol gro-Bog'k