There are a few things you need to know here, before you jump head long into your father's life.
1. Until you are 18, you do what your mother tells you that you can do. If your mother is truly abusive, contact CPS or the equivalent in your area to report it, or report it to a teacher, guidance counselor, pediatrician, or other mandatory reporter. Fair warning, this could result in you ending up in a foster home so if this is really more of a case of Mom just being "difficult" or you two not getting along famously, think long and hard whether or not you want to pull the trigger on this and report her as abusive. If your health and safety is at stake, do it!
2. Your father can not sign his rights away unless you were being adopted by someone else. In that case, he chose to not parent and let someone else fulfill that role. It might have been the best choice, but it was a choice. The only other way he could have lost his rights is if a judge took them away, but that would have been due to his own pretty severe conduct which placed you in jeopardy. He might have cleaned up his act since then, but it is exceedingly rare and you will want to be careful the tiger really has changed his stripes.
3. He could have voluntarily agreed not to pursue custody/ visitation, or pay child support. It doesn't speak well of him if he went this route, but it was a choice he could have made. At any time he could have changed his mind, and unless a judge truly had terminated his parental rights, he had the right at any point to pursue a relationship with you and allow you to get to know your siblings and step family. He elected not to do so. Even if your mother didn't want him to do so, and even if he had previously agreed he wouldn't be in your life, all he had to do was file in court to enforce his parental rights and he very likely would have been granted at least some form of custody or visitation. What I'm saying is, this isn't all on your mother, no matter how awful you seem to think mommy dearest is. She had virtually no say in it at the end of the day and any say he allowed her to have was his own choice.
4. This guy may not be a bad person, but he chose not to be part of your life to this point and he chose to have a family with others. Realize that going in. If it were a Hallmark movie he would realize what he missed out on and you would all become besties and life happily ever after Brady Bunch style, but that isn't always how it works out in real life. Hopefully you have a counselor or spiritual advisor you can turn to to help you sort through the joy that comes with family, especially those you haven't known your whole life.
