The railway station “Kaliningrad passazhirsky” (passenger station in “Yuzhny” railway terminal) provides communication with such cities as: Moscow, Saint Petersburg, Chelyabinsk, Adler, Murmansk, and others. Some destinations is provided only in summer.
Please check actual information on the website: rzd.ru (English menu)
To be allowed to go from Kaliningrad to other parts of Russia and back crossing the Lithuanian border, citizens of the Russian Federation have to be issued a special document – Facilitated Rail Transit Document (FRTD). In order to get it, a Russian citizen has to buy a ticket in any booking office in the Russian Federation not later than 28 hours before departure.
Russian citizens are not allowed to go by the train without a valid facilitated transit document or visa. The duration of stay is no longer than 6 hours. The Facilitated Rail Transit Document does not allow crossing the Lithuanian border by other means of transport. It is issued only for citizens of Russia.
Citizens of other countries have to get the transit information on their own: they have to be allowed to visit all EU and non-EU countries en route (e.g. Belarus).