The passport application process is very straight forward, just follow the instructions. The documents you need are clearly specified on the application and the instructions (both available from
http://travel.state.gov). If your son is now a US citizen (if he's under 18, in your custody and has a green card, then he likely is), you can apply for a passport for him.
However, since the state department requires original documents with the application, you can only apply for one passport at a time (both you and your son's citizenship is based on your naturalization certificate).
In addition to the documents on the passport instructions the state department asked us for our marriage certificate before they'd process my daughter's passport. This delayed the passport somewhat (a week or two), so you might consider adding it into the application package.
You can also apply for a citizenship certificate for your son using a USCIS N-600. Passports come fairly quickly, but N-600s come at the normal USCIS pace (the state department deals with citizens, the USCIS pretty much treats their "customers" like cattle).
Your son doesn't need a citizenship certificate (a passport is perfectly valid proof of citizenship) but some folks want the extra document (we did). This can be done in parallel; the USCIS does not demand original documents in the application, only copies. At the end of the process (typically about 6 months, I'm told) they call you (and your son) in for an "interview" where you show the original documents (this includes your son's green card, hold on to it).
We applied for an N-600 for my daughter at the end of June. We haven't heard anything (other than getting a receipt in the mail) since then. Amazingly, the N-600 process is even more opaque than the N-400 process -- the district office handles everything, so you don't get official "Notices of Action" (we got a handwritten receipt for the N-600!). As far as I know, there is no way to find out the status of an N-600 (there are ways that sort of work sometimes for N-400s). During my interview last month, I asked the IO about my daughter's N-600 and she said there was no way for her to find out what was happening.
I have no idea about the implications for Indians (Canada allows dual citizenship, so I'll be all set once I take the oath about a week from now (my wife took her oath in June)).
Congrats!
