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My daughter was born on Dec 2nd, 2008 in Austin, Texas. Here I am telling what happened for us in TX. It might be little different in your state.
After the delivery, the hospital itself give you the papers for Birth Certificate, SSN and Vaccination record. The documents are supposed to be fill by the mother. You should give clear details in all aspects. Especially baby's name.
We got SSN after two weeks from the date of her birth. The hospital instructed us to apply for Birth certificte after three/four weeks. You can either apply it in person or through online. The hospital itself given the address where to apply. We applied through Online and got it in next 10 days. (Actually 5/7 working days). For Birth certificates they give three different sizes like Full, Walltet etc., you can select as per your need and also the number of copies. For each copy we paid around $22.
Regarding the passport, after getting the Birth certificate you can apply at any time. Passport application form can be downloaded / get it from post office and filled. But dont sign in the application form prior. Check the post office having the passport application facility and get the appointment. Both mother and father should appear in person with their pasports and visa. After the verification they ask you to sign in front of them. Regarding the photo you can take it in the post office itself. We paid $101 for passport process (including photo for $15). Better if you can take it in home and print it. Even its very cheap if you print in Walmart (less than a dollar for 8 copies). In post office they never took good snaps. We got our daughter's passport in two weeks by USPS mail. If you are in hurry you can get it in a week by paying additional $60. Along with the passport application form you have to give the original birth certificate for verificarion. They will return it in a seperate mail after one week when you get the pasport.
I think this information is more enough for you, if you have any doubts ask me, I can help you...