Blue Magic

Spells able to be learned by a Blue Mage.

Goblin Punch

 * Physical attack.
 * Punch repeatedly in the direction you are facing.

Where to Learn

 * Any goblin type enemy, Leggi Steppe is a good location

Fire Breath

 * Magical attack: fire.
 * Expel flaming breath in the direction you are facing.

Where to Learn

 * Diabolos cave (Demon); Leggi Steppe & Hibat Foothills (Lizard); Quest (Red Dragon)

Leaf Cutter

 * Magical attack: wind.
 * Conjure a twister of leaves in front of you capable of dealing multiple hits.

Where to Learn

 * Mandragoras, Lake Tilouz

Aqua Breath

 * Magical attack: water.
 * Shoot three unpredictable bubbles straight ahead. The bubbles burst on contact.

Where to Learn

 * Forfex Fighter (purple crabs), Maxon Woods

1,000 Needles

 * Launch a blur of needles straight ahead that deals exactly 1,000 damage.
 * 2,000 damage for critical hits.

Where to Learn

 * Any cactuar type enemy, Debbis Heights is a good location early on

Aero

 * Magical attack: wind.
 * Whip up a small gale and launch it straight ahead.

Where to Learn

 * Any ahriman type enemy

Aerora

 * Magical attack: wind.

Where to Learn

 * Ramuh (★★★★)

Aeroga

 * Magical attack: wind.
 * Whip up a huge gale and launch it straight ahead.

Where to Learn

 * Ramuh (★★★★), high level Ahriman

Bad Breath

 * Produce a revolting stench that can potentially inflict poison, sleep, blind, silence, fog, and confuse on enemies.

Where to Learn

 * Any Malboro type enemy, Maxon Ravine is a good place early on.

Grand Delta

 * Magical attack. Generate a triangle of energy in front of you that bypasses enemy defense.

Where to Learn

 * Magic Pots, Hibat Caverns (resonance spawned)

Where to Learn

 * Behemoth, Quest (★★★★★★★), random spawn in Hibat Caverns

Where to Learn

 * White Chocobos, Quest "Black and White" (★)

Mighty Guard

 * Increases the defense of allies within range and causes them to float

Where to Learn

 * Iron Giant, Omega Weapon, Quest "Test for Alchemist and Geomancer" (★★★)

Much credit goes to /u/micahdraws for his Blue Magic Learning Guide thread on Reddit