You would need to know the full names of your mother and father. When and where they were married would also help. Where did they live when you were born? That would be the state where you would apply to the Division of Vital Statistics to determine if you have full brothers and sisters. You can also apply there to see if your father remarried. Normally it is best in cases like this to go to the Vital Records Office in person with whatever information you have (birth certificate, picture ID), and talk to someone. Birth records and marriage records aren't released to the general public, so you would need to prove you were a child of this couple to learn anything.
Please let me know if you have questions.