These achievements are awarded for beating all 20 levels of each world. Beating a level just requires you to reach the finish, and nothing more. This can either be done in time trial or star collecting mode, but it is suggested you stick to time trial mode as it is easier to get the gold medals. The first level of each world is unlocked from the start. You must beat the level to unlock the subsequent levels within each world. These achievements will unlock along the way to earning all of the gold medals.