To skip levels, enter the Level Select code and start the game on any level. To skip levels from there, pause the game and press A. If done correctly, you will be taken directly to the next level.
To get to the level select mode, go to the title screen (where press start is displayed) and press B,A, Right, A, C, Up, Down, A. If done correctly, you will be taken to the main screen. Press the start option and you will be taken to the level select mode.