While playing a level, press Z, L, R, and A at the same time to skip to the next stage.
At the Puzzle Mode screen (were you select A or B) press Left, Left, Up, Down, L, R, L, R, and then L&R at the same time. This will bring up a character select screen, where you can select either color dino, or one of the bosses. Be sure to press each button slowly, try not to go too fast.
At the main menu screen that says, "Game Start", Time Attack", and "Options", enter the following code: L-Button, up on D-pad, R-Button, and down on D-pad. This will allow you to reach "another world" full of new puzzles. After entering the code, a tiny character will appear in the lower right portion of the screen. Select "game start" and when you select "Puzzle Game", it will say another world underneath it. Press A and you'll go to another world.