Indian passports shuold be cancelled once you take the oath as a US citizen. Indian passport should not be used for travel to India following the oath ceremony.
Most people cancel Indian passport when they apply for OCI and send the passport along with the OCI application but that is not a must. If you are applying for OCI soon after getting US passport you can send the OCI application without Indian passport and then when you get the OCI documents at your consulate you can send your US passport for OCI stamping and Indian passport for cancellation. This is what I did.
Again depending on your consulate(NY,DC,Houston,Chicago) you may want decide when you should cancel your Indian passport because New York consulate turnaround is very quick(3 to 4 weeks) compared to other consulates so you can hold on to your Indian passport till you get your OCI if are in NY jurisdiction.
Hope this helps.