At least one parent must be a US citizen at the time of the baby's birth, AND have met residency requirements in the US to be eligible to pass on that citizenship to offspring. Both parents must go with baby in person to the nearest US consulate to place of the birth to register birth to US citizen(s) while abroad. You need baby's long-form birth certificate (which includes both parents' citizenships), hospital records, passports for each parent, proof of years of residency in the US for US-citizen spouse, baby's passport photos. If parents were not married at the time of baby's birth, DNA tests required all around to prove paternity. Both parents must consent to baby obtaining a US passport.
You'll need baby's US passport to bring baby back to US. Under US law all US citizens can only enter/exit US on their US passport. Penalties apply for failure to do so. Besides, entry on foreign passport means visa/visa waiver required & baby has to depart US. No change of status from tourist, so baby MUST have that US passport to enter if it is a US citizen.