Level Up Moves:
* Level 1: Scratch, Tackle
* Level 4: Ember
* Level 7: Quick Attack
* Level 10: Smokescreen
* Level 13: Growl
* Level 16: Flame Wheel
* Level 19: Tail Whip
* Level 22: Flamethrower
* Level 25: Defense Curl
* Level 28: Swift
* Level 31: Fire Spin
* Level 34: Incinerate
* Level 37: Will-O-Wisp
* Level 40: Overheat
TM/HM Moves:
* TM10: Hidden Power
* TM11: Sunny Day
* TM18: Rain Dance
* TM21: Frustration
* TM22: Solar Beam
* TM27: Return
* TM29: Psychic
* TM32: Double Team
* TM38: Fire Blast
* TM43: Secret Power
* TM44: Reflect
* TM45: Attract
* TM48: Skill Swap
* TM50: Overheat
* TM52: Focus Blast
* TM53: Energy Ball
* TM67: Facade
* TM87: Swagger
* TM90: Substitute
* HM01: Cut
* HM04: Strength
* HM05: Flash
* HM06: Rock Smash
Other Moves:
* Egg Move: Quick Attack
Important Notes:
* Evolution: Cyndaquil evolves into Quilava at Level 14 and then into Typhlosion at Level 36. Evolution unlocks new moves for your partner Pokémon.
* Move Tutor: Some moves can be learned by visiting the Move Tutor, but it's not clear from your question if you want that list too.
* Compatibility: Cyndaquil might learn different moves in other Pokémon Mystery Dungeon games.
Let me know if you'd like more information about any of these moves!