It is possible for Weavile to learn the following moves at the corresponding levels:
Level 8 - Quick Attack
Level 10 - Screech
Level 14 - Faint Attack
Level 21 - Fury Swipes
Level 24 - Nasty Plot
Level 28 - Icy Wind
Level 35 - Night Slash
Level 38 - Fling
Level 42 - Metal Claw
Level 49 - Dark Pulse
Weavile can also learn a number of moves available through using a TM or HM.
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If you breed it with a Ditto or another Weavile, (of the opposite gender) it should start to learn moves.
It should already know Growl and Charm, but it learns Metronome at level 5.
The way a Weavile can learn Ice Punch is probably if you teach it. Ice Punch isn't a TM so you have to go to Pastoria City and talk to the Move Teacher guy. You have to give him a heart scale in return( you can find heart scales underground).
Yes, it can learn any move, but it has a limited moveset by leveling up. It can learn a new move every 10 levels, but it can learn any TM or HM available. It can also learn all the tutor attacks. This is a great Pokemon to have to build an ultimate moveset.
the attacks my togetic learned were swift, metronome, and growl