Departing from Leningradskiy Station
Trains from Moscow to Estonian capital depart from Leningradskiy station located on Komsomolskaya Square (which also boasts two more railway stations). The railway station was designed by Konstantin Thon between 1844 and 1851. The architect has designed many famous buildings in Russia including the Cathedral of Christ the Savior and the Grand Kremlin Palace in Moscow.
The station has had different names in the past. Firstly it was named Peterburgsky station then to honour the Emperor after his death - Nikolayevsky and after Bolsheviks era - Oktyabrsky. Now it also called Moscow - Passazhirskaya.
Take a look around the Railway Station in Moscow.