Question: We are traveling from Tbilisi to Ataturk airport and then with a transit to Antalya; both legs were sold under a single ticket by Turkish Airlines. 

* Do they automatically put luggage for transit?
* How far is the second terminal, and where we should go for passport control?

Answer: 
For International to Domestic connections on Turkish Airlines, your bags will be automatically transferred to the domestic flight IF (and only if!) the domestic destination you are travelling to has customs facilities at the airport.

In the case of Antalya, they DO have such facilities, so your bags will be automatically transferred to your domestic flight.

Shortly before landing into Istanbul an announcement will be made on the flight, and they will direct you to the back of the in-flight SkyLife magazine to confirm which locations do and do not have customs facilities, so you can use this to confirm what I've said above.

Upon landing in Istanbul you will need to proceed through Passport Control (just follow the signs), through baggage claim and customs (WITHOUT collecting your bags), and then make an immediate LEFT turn and follow the signs to the domestic terminal which is a 5-10 minute walk away.
* I just always ask about luggage at the check-in when I drop it off to be extra sure. I expect that it would be transferred automatically in your case.
* There is no way that you will miss the passport control (just follow the signs), and Ataturk is a single large building so different places in it can't be very far.