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Modern Subway Station in the Tbilisi International Airport takes you in 25 minutes to Tbilisi CentralRailway Station (called "Vagzlis Moedani"). Train Departure shedule tothe City every day at: 01:45; 5:45; 9:00; 11:45; 13:10; 18:15; 21:15
Airport train will get you to "Vagzlis Moedani" and then you can use Metro, or Busses. Here is the link of Tbilisi Metro plan. Tbilisi central railway station "Vagzlis Moedani" is connected with other Georgian, Armenian and Azerbaijan cities, this is link of Georgian Railway.
Please find the info about Tbilisi bus routes at Info-Tbilisi