You must have a gamecube memory card in order to do that. There should be a slot for that on top of the Wii if it is a version old enough that can run gamecube games.
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To save your game, you click on your diary and press save. Its on a bedside table it should have a lamp on it.
you can play the game cube version on the Wii, but there is no super Mario sunshine 2.
Harvest Moon Magical Melody; available for Game-cube and Wii.
How does one save their progress on Petz Crazy Monkeyz game disc (Wii)?
You don't. After you complete a mission it will save by itself.