You need to fill out an application for CRBA and passport, bring yourself/ kiddo/mom to the embassy, and apply. One of the documents we needed was the translated foreign birth certificate and I needed proof of 5 years in the US. Used college transcripts, old passports, and tax info.
When you apply, you choose a name. No name selected matters up to that point LOL.
The two applications are submitted together. You aren’t dealing with the immigration side, you’re dealing with ACS (American Citizen Services) whose only job in the world is to make these things easy for you to do.
If you want, and they will ask you, you can apply for the SS card at the same time. It’s good to have to enroll your baby in health insurance etc.