Can I travel by rail from Turkey through Georgia to Russia ?
Currently (June 2014), there is no rail link between Turkey and Georgia.
A railway link from Turkey to Georgia is under construction: see Kars-Tblisi-Baku railway on Wikipedia.
The railway link between Georgia and Russia is currently closed due to the political conflict in Abkhazia. See Georgian Railways, again on Wikipedia.
You might be able to complete your journey if you do part of it by bus. You might also be able to travel from Georgia to Azerbaijan and from Azerbaijan to Russia. However, Wikivoyage reports that the Russian-Azeri border is closed to non-CIS-passport-holders.