I think that it will be quite difficult to trace your mother's birth family if she was adopted, as the records are likely only to be available to her and her mother.
Have you thought about trying to trace her, or some of her adoptive family if any are still alive?
If your parents were married have you thought about tracking down their marriage certificate?
To trace your father's family you need his birth certificate. That will give you his parents names, and then you can search for siblings by inputting his father's last name and his mother's maiden name.
If you want contact with any living relatives you might want to think about searching social media - Facebook etc.