To be succinct, the answer to the question is "NONE", without a court order. Even if the father is designated as such on the birth certiﬁcate, the father has no enforceable rights. The child's mother is completely within her legal rights to refuse to allow the father to even see the child, let allow visitation or care. This is also true for the father's entire family, so that any grandparents, uncles, aunts, cousins, etc., would also be unable to have any access without the mother's express permission. However, even if that permission were to be granted, it can be withdrawn arbitrarily at a moment's notice.
Why is this more relevant today? Statistics show the marriage as an institution has been on a steady decline for decades. Furthermore, when couples do marry, they are considerably older than couples of a generation ago. Because of these trends, the incidence of couples living together has exploded. This, in turn, has led to a dramatic increase in the number of children born to couples out of wedlock.
For father's who may have moved to Texas from another state which may or may not have similar paternity statutes, Texas law will prevail after the child has lived in Texas 6 months.
What are the implications of this? While an unmarried father may have had routine access and visitation to their child for years, the child's mother can legally pick up and move to another state or another country without any notiﬁcation of the child's father.
An unmarried father in Texas who is interested in insuring a role in their child's life
should see an attorney promptly to legally establish his parental rights. When a judge issues a ﬁnding of paternity, the father will have the same rights as a divorced father would have, including, for example, conservator rights (the right to participate in important decisions such as where the child lives, educational decisions, medical decisions, etc.), visitation rights, even custody.