My wife and I are both foreigners living in Korea and we're expecting a baby?

Congratulations. The key thing to understand first is that Korea grants citizenship by bloodline, not by place of birth, so a baby born in Korea to two foreign parents does not automatically become a Korean citizen. Your child takes your nationality, which means the main registration you do is with your own country's government. In practice, the hospital gives you a birth certificate (출생증명서), and you then register the birth with your home country's embassy or consulate in Korea to establish the baby's citizenship and get their first passport; each embassy has its own list, but generally the hospital certificate (often needing a translation and sometimes an apostille), both parents' passports, and your marriage certificate.

For the baby's stay in Korea, you must report the birth to Korean immigration and get the child a visa status and Alien Registration Card, usually within 90 days of birth, typically as a dependent matching the parents' status; bring the birth certificate, passports, and your ARCs to your local immigration office (book on Hi Korea). It is also smart to enroll the baby in National Health Insurance and notify your employer for any dependent benefits. So the order is: hospital birth certificate, then your embassy for citizenship and passport, then Korean immigration for the child's ARC. The 1345 hotline and your embassy can both confirm the exact papers, and starting the passport process early helps since it can take a few weeks.