Successfully complete World 1 to unlock an option for the Staff Credits mini-game at the Options menu.
Successfully complete all eight default worlds without using any continues to earn a crown in each of them to unlock the ninth world, Sinking Swamp.
Successfully complete World 9 without using any continues to unlock World 10.
Successfully complete World 1 through World 9 with only one character (either AiAi, Baby, Doctor, GonGon, MiMi, or YanYan) to unlock the "Super" version of that character. The "Super" version of that monkey can accelerate slightly faster.
Successfully complete all mini-games at least one time. Then, play any mini-game to force the game to save your stats. Then, highlight Baby at the character selection screen of the main game and press 1 to play as Baby Robo from the mini-games.