Mew learns a new move exactly every 10 levels and, of course, you can teach it TMs, HMs and Move Tutor moves as well.
If you'd like a detailed list, the related link below is very helpful.
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Level when you get it
Mew's SpecialtyMew, an extraordinary Pokemon, can virtually learn all moves no matter what. However, if you try in the Pokemon games, Mew can't possibly learn special moves like Blast Burn, Frenzy Plant, Hydro Cannon, Volt Tackle, Heart Swap, Judgement, etc. In FireRed and LeafGreen, the lady at Cape Brink only teaches elemental Hyper Beams to Venasaur, Blastoise, and Charizard. In the games, Mew can only learn the moves that it learns by leveling up, from Move Tutors, and all the TMs and HMs that you have.
Here are the moves magikarp can learn. Any level - Splash Level 15 - Tackle Level 30 - Flail Evolves into gyrados at level 20
Level 13
the moves charmander learns go as: level 7- ember, level 13- metal claw