You have no parental rights while she is pregnant because until the baby is born, you have no way to prove that you are the child's father. They will not let you be present at birth without her permission, and you cannot sign the birth certificate without her permission.
After the baby is born, you can petition the family court in whatever county she lives in to establish paternity. There will be a bunch of papers to fill out, and she will need to be served, then those papers will get returned to court and you'll get a court date. At that court date, they would order that she allow a DNA test to establish paternity.
Once that test comes back, if you are the father, then your legal rights and responsibilities begin. After you are judged by court to be the child's father, there will be another court date to establish custody and child support. Given that you'll be talking about an infant and live far away, it would probably be best to assume that the mother would be the custodial parent and that you would pay child support and have visitation. If you move closer and it's in the best interests of the child for you to me more involved, you can figure that out later.
She's probably pretty freaked out right now. Give her time and space but let her know that you want to support her and be there for your child, with or without a relationship with her. I would check in every now and again but keep it low key and no pressure. She's re-evaluating her life now in light of the fact that everything is changing and is focusing on how she's going to support herself and a child. That's a pretty big deal. You're obviously freaking out too, but she's also the one dealing with the physical, hormonal aspect of pregnancy.
If I were you, I wouldn't plan on any major life changes until the baby is born and you know it's yours. That's a really important factor in this - I'm sure you're sure it's yours but until you have a DNA test, you really don't know. Then if you really are the father, you'll have to figure out whether or not you're going to try to be a parent from really far away or will move closer to your child, even if you no longer have a relationship with your child's mom other than shared parenting. In the meantime, make sure you've got a good job, start saving money, live a clean life, and show that you are good father material.
FWIW, my husband and his daugther's mother broke up when she was pregnant. It was a pretty nasty break up but he was able to be there for the birth after the mother came around to realizing that he wasn't going anywhere. He started sending her money and visiting even before they had a court order in place, then got weekly visitation, etc.. She lived with her mom for 13 years and we actually won custody of her two years ago and she lives with us. You never know how things are going to turn out. So don't freak out - be supportive but not psycho, let her know that you're here for her and the baby without being harassing, and then when the baby is born, file the complaint for paternity and let the court process work.