When you become US Citizen, when & where do you have to surrender Indian passport? What if you need to travel before getting US passport or getting OCI?
Since india does not allow dual citizenship, you are no longer an indian citizen as soon as you take the USC oath. No one asks for you to surrender your Indian passport at US citizenship oath ceremony.
You get the Indian passport cancelled later by mailing it to the Indian embassy with proof of US citizenship (copy of citizenship certificate or copy of US passport). They send it back to you with the cancelled stamp.
Nevertheless, use of your indian passport for travel or any other citizenship related purpose is considered illegal as soon as you have taken USC oath. So dont even think of using it for travel.
You can apply for US passport immediately after getting oath (provided you have all the required documents ready). It takes anything from 2-5 weeks.
You can NOT travel out of US without passport
(Travel to Canada and Mexico used to be allowed with just the citizenship certificate, but I dont think it is any longer.)
If you are in a hurry, apply for the expedited US passport service, then go personally to indian embassy and get a tourist visa to india.
And accept the fact that there will be some days in this whole process where you CANNOT travel out of US.