In Texas, it is unlawful for a judge to favor mothers during a custody hearing. Ideally, this law recognizes that mothers are not inherently better parents than fathers. However, every case is different. You might need to fight harder than the mother to be granted custody rights and a fair amount of child support.
What factors influence a court's opinion?
Above all, a court will need to see that you have an active interest in the child's life and their best interest. They will want to see that you are prepared to be a parent. Their opinion will influence your custody and child support arrangement.
Many people in today's culture are inclined to believe that women are "naturally" inclined to be better caretakers than men. However, these might be based on subconscious bias rather than evidence. Not every woman is suited to be a good mother, and not every unmarried man is automatically an unfit father.
Importantly, the court might need to establish legal paternity. If you were not married to the mother or in a committed relationship when the child was conceived or born, you might have to provide additional evidence to prove you are prepared to be a father. Other than signing the birth certificate or a Recognition of Parentage, a court might ask that you take a paternity test. In contrast, it is rare that mothers need to take a maternity test.
In addition, if you were an active step-parent you may be able to establish paternity outside of biological responsibility. Those situations need careful legal attention, as some men may unknowingly be victims of paternity fraud.
How else is a father's situation different than a mothers'?
Fathers often provide a different role in a family than a mother, but your role is just as valid as a mother's. Therefore, your behavior and character cannot be judged in exactly the same way as the mother. A court will want to see that you acknowledge any past behavior (especially a criminal or violent past involving the mother).
Another cultural expectation that men often feel the need to fill is the role of the provider. It should be enough that you, like any parent, can provide for your child's emotional and physical needs. You should not face unfair bias due to your salary or your education level.
If you are concerned that a court will automatically favor your child's mother in your custody battle, you may need to explore your legal options.