By leveling up:
* Scratch: A basic physical move, learned at level 1.
* Growl: A move that lowers the opponent's Attack stat, learned at level 5.
* Ember: A Fire-type special move, learned at level 7.
* SmokeScreen: A move that creates a smokescreen, hiding the user, learned at level 13.
* Leer: A move that lowers the opponent's Defense stat, learned at level 18.
* Flamethrower: A powerful Fire-type special move, learned at level 25.
* Dragon Claw: A powerful Dragon-type physical move, learned at level 33 (Charmeleon only).
* Flame Wheel: A Fire-type physical move, learned at level 41 (Charmeleon only).
* Solar Beam: A powerful Grass-type special move, learned at level 49 (Charizard only).
* Fire Blast: A very powerful Fire-type special move, learned at level 57 (Charizard only).
By TM/HM:
* Quick Attack: A fast physical move.
* Facade: A powerful move that ignores the user's stat reductions.
* Thunderbolt: A powerful Electric-type special move.
* Thunder: A powerful Electric-type special move.
* Earthquake: A powerful Ground-type physical move.
* Aerial Ace: A fast Flying-type physical move.
* Flame Charge: A physical move that raises the user's Speed stat.
* Heat Wave: A powerful Fire-type special move.
* Overheat: A very powerful Fire-type special move, but lowers the user's Special Attack stat.
* Will-O-Wisp: A move that burns the opponent, lowering their Attack stat.
* Dragon Pulse: A powerful Dragon-type special move.
By breeding:
* Ancient Power: A powerful Rock-type special move.
* Blast Burn: A very powerful Fire-type special move, but it takes 2 turns to charge.
* Dragon Dance: A move that raises the user's Attack and Speed stat.
* Extreme Speed: A very fast physical move.
* Hidden Power: A move with varying type and power, depending on the Pokemon's individual IVs.
* Outrage: A powerful Dragon-type physical move, but the user becomes confused after using it.
* Swords Dance: A move that raises the user's Attack stat by 2 stages.
This is not an exhaustive list, as Charmander can also learn moves from other Pokemon via breeding and various other methods.
You can find a full list of moves that Charmander can learn by searching online for "Charmander move list" or "Charmeleon move list" (for Charmeleon) or "Charizard move list" (for Charizard).