Monk trainers

A monk trainer is an NPC that offers monks the opportunity to reset their talents and change their specialization. A general rule-of-thumb for finding the local Monk trainer is to search for either a peaceful, secluded area where one may be meditating, or in a local tavern putting their drinking skills to the test. Because the leaders of the respective Alliance and Horde Pandaren factions are Monk trainers themselves, keep an eye out for the turtle mounts and Pandaren architecture!

Alliance trainers

Location	Monk Trainers
Eastern Earthshrine, Stormwind City	Aysa Cloudsinger
Training Hall, Stormwind City	Juisheng Halfclaw Katrina Long
Northshire Abbey, Elwynn Forest	Bao
Stonefire Tavern, Ironforge	Liu of the Bottomless Tankard
Anvilmar, Dun Morogh	Lo
New Tinkertown, Dun Morogh	Xi, Friend to the Small
Temple Gardens, Darnassus	Lanfen, Brewer of Fine Teas
Aldrassil, Teldrassil	Laoxi
Ring of Arms, Exodar	Pan of the Martial Spirit
Ammen Vale, Azuremyst Isle	Mojo Stormstout

Horde trainers

Location	Monk Trainers
Valley of Honor, Orgrimmar	Ji Firepaw
Barracks, Orgrimmar	Huinli Wingpaw
Valley of Trials, Durotar	Gato
Echo Isles, Durotar	Zabrax
Spirit Rise, Thunder Bluff	Tseh Rah, Peacebloom Teamaster
Camp Narache, Mulgore	Shoyu
War Quarter, Undercity	Ao Jian
Deathknell, Tirisfal Glades	Ting, Strong of Stomach
Farstriders' Square, Silvermoon City	Sho the Wise
Sunstrider Isle, Eversong Woods	Pao

Neutral trainers

Location	Monk Trainers
Greenstone Village, Jade Forest	Brewmaster Tzu
Peak of Serenity, Kun-Lai Summit	Fearsome Jang Iron-Body Ponshu Number Nine Jia Wise Scholar Lianji
Temple of Five Dawns, Wandering Isle	Fearsome Jang Iron-Body Ponshu Number Nine Jia Wise Scholar Lianji Lao Li the Quiet

Trivia

The monk trainers are unique among class trainers in that they are almost exclusive pandaren; the first exception was the troll Zabrax, who has apparently been trained by the nearby Tsu the Wanderer, later followed by human monks Lao Li the Quiet and Katrina Long. In the case of the starting area trainers, this usually makes them the only class trainers which are not of that area's race. Before Shadowlands, they were also all level 90, despite the starting area trainers being found in areas where most other NPCs are somewhere around level 5.
Following the two pandaren groups' joining of the Alliance and the Horde, monk trainers traveled to each allied race's starting area to teach the ways of the monk, and also to learn the ways of the other races. These pandaren are distinctive characters, and in many cases have clearly chosen (or been chosen) to train that specific race due to their personalities. Some examples:

Gato is a well-armored pandaren sent to train the orcs.
Ting, Strong of Stomach is obviously well-chosen to handle life among the Forsaken. Her strong stomach impresses even Archibald Kava, when he puts it to the test with a drink of dubious safety.
From her title, Tseh Rah, Peacebloom Teamaster seems a fitting match for the nature-loving tauren.
Likewise, Xi, Friend to the Small seems a natural choice for training the gnomes.
Liu of the Bottomless Tankard finds ample opportunity to illustrate the origin of his name among the Ironforge dwarves in the tavern where he spends most of his time.