What level does mew learn its moves?

It can learn any TM or HM. And its Level Up Moveset is made up of the following:

  • At Lv. 10, Mega Punch
  • At Lv. 20, Metronome
  • At Lv. 30, Psychic
  • At Lv. 40, Barrier
  • At Lv. 50, Ancientpower
  • At Lv. 60, Amnesia
  • At Lv. 70, Me First
  • At Lv. 80, Baton Pass
  • At Lv. 90, Nasty Plot
  • At Lv. 100, Aura Sphere
So all in all, not a bad moveset, except it's probably best to avoid all the attacks learned after Lv. 50 other than Aura Sphere.