If what you want to do is have both your parenting rights and those of your child's father set by the court, you will need to start a paternity action.
Missouri has legal and physical custody. Physical custody is self-emplanatory. Legal custody involves a parent's right to be involved in decision making - for example, in making medical decision, involvement in school and extracurricular activities and communicating with the child.
The preference that is set out by the legislature is for parents to have joint legal and joint physical custody - which does not necessarily mean a 50/50 sharing of parenting time. That said, for very young children who have not had contact with the nonresidential parent is several months, most courts will set up a phase in for parenting time unless there is a really good reason to restrict parenting time.
If your child's paternity has not been administratively determined through a DNA test, the courts may want you to secure a DNA test at some point prior to rendering a judgment.
Answers given here are for general information only and do not create Ana attorney client relationship. The choice of an attorney is an important decision and should not be based on advertisement. I only am licensed in the state of Missouri. If you are seeking information in a different jurisdiction, you should seek an attorney in your jurisdiction.