What game are you trying to use it with? Make sure the game is compatible with the classic controller - or it will not work at all.
Did you make sure it was plugged into the Wiimote before you loaded the game?
I hope you get it working - if not then please give us more information. Thanks.
EDIT: For playing Gamecube games you will need a gamecube controller, which plug into ports under the flap on the side of the Wii (or the top if you have it standing up).
There are four controller ports and two memory card ports under that flap.
You will need a Gamecube controller to play gamecube games, and also a memory card if you want to save.
The classic controller that plugs into the wiimote is generally used for the download virtual console games from the Wii Shop Channel (as well as some wii games) but can't be used for Gamecube games I'm afraid.
Can you take it back to the store?
EDIT2: There IS a wireless gamecube pad, one part plugs into the ports under the flap and the pad is wireless and connects to that. The pad you want is called a WAVEBIRD.
I think they are pretty tough to get hold of now but we got one a year ago for our gamecube and use it on the Wii now with no problems and it's a great pad.
I hope you manage to get one. Good luck. :)