You will need a lawyer to attend to the surrogate relinquishing her parental rights in your favour. In some US states that can be arranged in advance of the birth (or, in California for example, you can get a pre-birth judgment placing both dads on the birth certificate.) Surrogacy agencies will normally provide a referral if required.
You should seek legal advice before entering a surrogacy arrangement, as the laws vary widely between different US states as to prohibitions on compensated and uncompensated surrogacy.
In some starts, courts will enforce gestational surrogacy contracts, in others they will not.
Some states have laws protecting surrogacy, others rely on statues or published cases to guide legal practice.
In some states parental birth orders are routinely granted in the absence of specific law or statues. In others this is unpredictable.