Archery (Women) team event is on July 29, and that may be India's first realistic Gold prospect. Deepika Kumari is ranked #1 in the world. So chances of individual Gold are stronger than a team Gold, but for that you will have to wait till August 2.
I agree with the Anon user that registering a double digit Gold medal is an unrealistic expectation, but expect this to be India's best ever Olympics both in terms of Gold and total medal tally.
Some events where I am hopeful:
- Tennis (At least mixed Doubles)
I hope Hockey team manages to surprise by getting a Bronze, but that may be being a bit over-optimisitic.
See question on Quora