Here is my Quicklist Guide to the Alliance Side of the Pet Trainer Intro Quests. This is a quick reference to see what type of pets you will need to defeat the trainers easily. On Trainers who have the same type of pet for all 3 pets in their party, I would recommend you bring at least two pets that are the opposite type (i.e. bring three appropriate leveled Mechanical types to fight three trainer Beasts). Don't rely on fighting all three of their pets with just one of yours, despite the fact that they will have stronger attacks and defeat them quicker.
1st Tier Alliance Trainer Questline (Starts in Eastern Kingdoms)
To defeat these trainers with maximum efficiency, you will need pets of the following type:
Mechanical, Beast, Elemental, Magic, Dragonkin, Undead
Julia Stevens - Elwynn Forest, Maclure Vineyards
Quickteam List-Minimum level 2 Mechanical Type
Pet 1: Fangs - Level 2 Beast - Use Mechanical
Pet 2: Slither-Level 2 Beast - Use Mechanical
Old MacDonald-Westfall
Quickteam List-Minimum Level 3 Beast, Elemental, Magic
Pet 1: Teensy-Level 3 Critter-Use Beast
Pet 2: Foe Reaper 800-Level 3 Mechanical-Use Elemental
Pet 3: Clucks-Level 3 Flying-Use Magic
Lindsay - Redridge Mountains
Quickteam List-Minimum Level 5 Beast
Pet 1: Flipsy - Level 5 Critter-Use Beast
Pet 2: Dipsy-Level 5 Critter-Use Beast
Pet 3: Flufftail-Level 5 Critter-Use Beast
Eric Davidson-Duskwood
Quickteam List-Minimum Level 7 Mechanical
Pet 1: Darkwidow - Level 7 Beast-Use Mechanical
Pet 2: Webwinder - Level 7 Beast-Use Mechanical
Pet 3: Blackfang - Level 7 Beast-Use Mechanical
Steven Lisbane - Northern Stranglethorn
Quickteam List-Minimum Level 9 Mechanical, Dragonkin
Pet 1: Moonstalker-Level 9 Beast-Use Mechanical
Pet 2: Nanners-Level 9 Beast-Use Mechanical
Pet 3: Emeralda - Level 9 Magic-Use Dragonkin
Bill Buckler - Cape of Stranglethorn
Quickteam List-Minimum Level 11 Magic, Undead
Pet 1: Eyegouger-Level 11 Flying-Use Magic
Pet 2: Young Beaky-Level 11 Flying-Use Magic
Pet 3: Burgle-Level 11 Humanoid-Use Undead
*Author's Note: In writing this guide, I used various references from wowhead and warcraftpets.com. I did not write this guide as I was doing each trainer. Some information might be wrong. Please feel free to comment and I will change it as needed.