The Wii can play GameCube games, but only the regular GameCube retail discs, you can't download them. And to play them you need a GameCube controller and memory card.
The Wii Shopping channel has games from older classic consoles like the NES, Super NES, Genesis, etc. They can only be purchased from the Wii Shopping Channel, not on your PC.
Out of curiosity, which games did you buy with it?