I am assuming you have schengen visa and also visa to enter Georgia. I don't think you need transit visa at istanbul airport, if you are just changing planes.
To answer to your question whether you need transit visa in Istanbul: http://www.mfa.gov.tr/frequently-asked-questions.en.mfa
Its says there:
Question: I will be flying to Europe. I know that our aircraft will
land in İstanbul. I am not planning to leave the transit lounge. Do I
need to get transit visa?
Answer: If you will not leave the transit lounge at the airport you
are not required to have transit visa. Otherwise, you have to make
visa application to the nearest Turkish Representation. Contact
information of the said missions can be reached through www.mfa.gov.tr
Also, refer to this official page: http://www.mfa.gov.tr/visa-information-for-foreigners.en.mfa
It says this:
India: Diplomatic passport holders are exempt from visa for their
travels to Turkey up to 90 days. Ordinary, Special and Service
passport holders are required to have visa to enter Turkey. Ordinary,
Special and Service passport holders with a valid Schengen or OECD
member's visa or residence permit may get their single entry e-Visas
valid for one month via the website www.evisa.gov.tr, provided that
they meet certain conditions.
If you want to enter istanbul, just apply for e-visa online to enter istanbul. Remember you collect baggage ONLY after passport control.
I am also indian passport holder, entered through istanbul airport several times needed no transit visa, to change planes to another destination within the terminal.