Battle Pet Tamers - Eastern Kingdoms

There are five pet tamers in the Eastern Kingdoms that are part of the quest Battle Pet Tamers: Eastern Kingdoms and these tamers use a variety of different pets which range in levels from 13 to 17 and are spread out all across the continent from Eastern Plaguelands at the top of the continent to the Swamp of Sorrows way down south.

It's worth mentioning that the way Blizzard has designed these quests, it's best to handle the Eastern Kingdoms tamers with an Alliance character because of the strange way the achievements are designed. Some players have reported that doing these quests and tamers on their Horde characters haven't allowed them to progress through the pet tamer quests correctly nor have they been awarded the right achievements after beating those tamers. Note this isn't something that happens all the time, but has been reported nevertheless.

David Kosse

David Kosse hangs out in the Hinterlands

Kosse is located in the Hinterlands and has a Beast named Corpsefeeder, a Magic pet named Plop and a Critter named Subject 142. Should your pets be low levels and around level 13 as Kosse's pets are, you would most easily beat him by utilizing: 

A pet with Mechanical attacks to beat Corpsefeeder the Maggot.
A pet with Beast attacks to beat Subject 142 the Rabbit. 
A pet with Dragonkin abilities to beat Plop the Ooze.

Kosse's general location is south of Stormfeather Outpost near the eastern side of the Hinterlands.

Deiza Plaguehorn

Deiza Plaguehorn sets up shop in Eastern Plaguelands

Plaguehorn is located in Eastern Plaguelands and has a Beast named Bleakspinner, another Beast named Carrion and an Undead named Plaguebringer. Her pets are all level 14. 

Should your pets be low level as well, you would likely beat her combination best with:

A pet that has Mechanical attacks to beat Bleakspinner the Spider.
A second pet with Mechanical attacks to beat Carrion the Maggot.
A pet with Critter attacks to beat Plaguebringer the Infected Fawn. 

Note that since her last pet is an Undead pet, it will resurrect after death and be immortal for one round, so plan accordingly and don't let your pets run out of HP before hers do!

Plaguehorn's approximate location is just west of Light's Hope Chapel across the lake.

Kortas Darkhammer

Kortas Darkhammer lives in Searing Gorge

Kortas is located in Searing Gorge and has three Dragonkin named Veridia, Garnestrasz and Obsidion. They are all level 15. If your pets are low level or you wish to level up on this tamer, a few pets with Humanoid abilities would likely do very well. Alternatively, pets that might have high resistance to magical damage (since Darkhammer's pets cast magic damage for some of their main attacks) would do well also.

Kortas's general location is right inside Thorium Point along the northern edge of Searing Gorge.

Everessa

Everessa enjoys the Swamp of Sorrows

Everessa is located in the Swamp of Sorrows and has a Flyer named Dampwing, an Aquatic named Croaker and a Beast named Anklor. All of her pets are level 16. Should you wish to level up your pets on this tamer:

A pet with Magic abilities would work well against Dampwing the Moth.
A pet with Flying abilities (likely a Flyer or a Dragonkin) would do well against Croaker the Frog.
A Mechanical pet or pet with Mechanical abilities would fare well against Anklor.

Everessa is located right outside the small Alliance base that sits on the northern shore of the lake surrounding the Temple of Atal'Hakkar and is just east of Marshtide Watch.

Durin Darkhammer

Durin Darkhammer hangs out in Burning Steppes

Durin is located in Burning Steppes and has a flyer named Comet, a Critter named Ignious and an Elemental named Moltar. All of his pets are level 17. 

Should you want to level up your pets on this trainer you'll want to use:

A pet that uses Magic abilities that would do well against Comet the Flame Fly.
A pet with Beast abilities that would work against Ignious the Beetle.
A pet that uses Aquatic abilities that would fight well against Moltar the Crab (he is apparently an elemental crab...).

Should you fare well against each of these tamers, your next pet battle on the Eastern Kingdoms continent would be against Grand Master Lydia Accoste in Deadwind Pass, which is also home to the coolest undead pet in the game, the Restless Shadeling.