Can I use apple cider from the supermarket to make hard or sweet cider?
No. Cider purchased at the super market will almost always contain preservatives and will not allow the juice to start fermenting. State and local laws require producers to treat apples with anti bacterial sprays prior to pressing and many require that stabilizers be added prior to sale.
Go to your local apple orchard and ask them for wine or hard cider grade juice. Arrange to pick up your cider the day you will start to make it. The fresher the better! You can also use frozen apple concentrate if you want. Just make sure to check the label for any preservatives. A good rule of thumb is if you see anything listed after apples in the ingredients, the juice won't work for making cider.