The most important thing you need to do is be sure to get your name on the birth certificate.
In Britain: Age doesn't play a part. If you're married, both parents automatically get parental responsibility. If not, the father only gets responsibility (rights) if his name is on the birth certificate. For this to happen, he has to be present when the birth certificate is made. If this doesn't happen the he'd have to contest in court at a later date and they'd probably make him pay for DNA tests on top of his legal fees.
As far as custody goes. If both parents have responsibility but are not together, it is almost always the case that a baby lives with the mother for the majority of the time and then the father has the rest (say, every other weekend and Wednesdays, or every weekend. Whatever you work out between you). If you have an arrangement between you, it can be more flexible but there is nothing official saying that she has to give you the child then or else. If you get something on paper then you're both legally committed to that arrangement.
In the UK, parental responsibility includes the right to make decisions about the child's religion, education and medical issues and to see the child, along with the responsibility to pay child support.
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I had a look at how it is in Kentucky and it's pretty much the same.
In the states, custody is divided into two categories. Legal custody and physical custody. Legal custody is having parental rights (what we call parental responsibility), physical custody is having the child under your care.
You have legal custody once you are on the birth certificate. The mother will sometimes have full physical custody but you will always have visitation rights (the right to see your baby). As the baby gets older custody can be shared more evenly (tip: it's super important to stay on your ex's good side because she will fight a lot harder against shared custody if she has a problem with you/doesn't trust you with the baby).
The only time you wouldn't get legal custody if you were unfit (this has to be something extreme like being a drug addict or putting the child in danger of abuse).
Once you've got legal custody by being on the birth certificate, you've jumped the biggest hurdle. After that, you can sort out between yourselves how much time you each get with the baby, when and how you divide the costs (e.g. child support payments). If one or both of you has a problem with the arrangements, you can go to court. This starts with a mediation, where you both sit down with a court appointed person who will help you discuss what you both want. If you agree to an arrangement here, it'll be put down on paper, become official and that's that. If you can't agree then you end up in family court where the judge will decide.
These links are really good:
Right of a Teenage Father and
Father's Rights Laws in Kentucky.
I'm excited but very nervous.
So far so good on staying on her good side, and certainly nothing I can think of that might make me unfit to get custody. She says she has no problem with me seeing the baby a few times a week, so hopefully she won't change her mind and some of the visitation drama can be avoided. I'm able and more than willing to pay child support also so I'm glad that that's something that we can work out ourselves. 
I'm very excited because in a few weeks we should find out if it's a boy or a girl. 
