Moscow to Kaliningrad by train.

Yes, the train runs through Belarus and Lithuania, so you will need the appropriate visas for those two countries (depending on your nationality).
You'll also need to have at least a double-entry Russian visa, as you will be exiting and then re-entering Russia.
Note that there is no immigration check on the Russian-Belarus border, but you still need a Belarus visa and when you try to exit from Belarus to Lithuania the Belarus officers will be most unhappy if you have no visa.

Thank you very much for detailed information.

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