-( Story related, cannot be missed )-
After concluding the trial from Chapter One, the Monokubs and Monokuma will give you four seemingly useless items. These items are heralded as treasures by the gifters and you will use these items to unlock new areas within the academy. Each corresponds to a different location among the academy and you will need to unlock them in order to proceed further through the story anyway.
The orb is used on the second floor before reaching the Ultimate Pianist's Lab by looking at the dragon statue. This will unlock the labs for Maki, Kirumi, Gonta and Ryoma.
The crank is used outside (diagonally to the right of the school building) on the main outdoor screen. If you go east of the hyacinth trellis, you will use the crank to open the casino and love building.
The passport is used on the first floor as if heading to the gym. After entering the partitioned section, the wall to the left will use the item and unlock the Ultimate Mage's Lab.
The ocarina is used outside of the school. Upon exiting, turn left and follow the path until you get to a wall of vines. The small statue to the left will have some music notes to hint at your use of this item. Using it will unlock the Pool.
All you have to do is use any one of the items to expand your school.