By leveling up:
* 1: Tackle
* 4: Bite
* 7: Dragon Rage
* 10: Twister
* 13: Scary Face
* 16: Hydro Pump
* 19: Rain Dance
* 22: Crunch
* 25: Ice Fang
* 28: Outrage
* 31: Dragon Dance
* 34: Aqua Tail
* 37: Earthquake
* 40: Hyper Beam
* 43: Bounce
* 46: (No move learned)
* 49: (No move learned)
* 52: (No move learned)
* 55: (No move learned)
* 58: (No move learned)
* 61: (No move learned)
* 64: (No move learned)
* 67: (No move learned)
* 70: (No move learned)
* 73: (No move learned)
* 76: (No move learned)
* 79: (No move learned)
* 82: (No move learned)
* 85: (No move learned)
* 88: (No move learned)
* 91: (No move learned)
* 94: (No move learned)
* 97: (No move learned)
* 100: (No move learned)
By TM:
* TM01: Focus Blast
* TM06: Toxic
* TM09: Bulk Up
* TM10: Hidden Power
* TM11: Sunny Day
* TM12: Taunt
* TM14: Thunderbolt
* TM17: Protect
* TM18: Rain Dance
* TM21: Frustration
* TM22: Solar Beam
* TM24: Thunder
* TM27: Return
* TM30: Shadow Claw
* TM32: Double Team
* TM37: Sandstorm
* TM39: Rock Tomb
* TM40: Aerial Ace
* TM41: Roar
* TM42: Facade
* TM44: Rest
* TM45: Attract
* TM48: Skill Swap
* TM52: Focus Blast
* TM58: Sky Attack
* TM67: Substitute
By breeding:
* Egg Move 1: Dragon Pulse
* Egg Move 2: Ice Beam
By interacting with certain Pokémon:
* Magikarp: (No move learned)
By using the Move Tutor:
* Hydro Cannon
* Ice Shard
* Dragon Claw
* Crunch
* Body Slam
* Iron Head
* Superpower
* Earthquake
This list may not include all possible moves Gyarados can learn, but it should provide a good overview of its movepool in Pokémon Emerald.