You can reach most of the cities in Siberia via flight or rail. Trans-Siberian Railway is the most popular mode of transport to Siberia to and from various parts of Russia, and even China and Mangolia.
No direct flights are available to reach Siberia from India. You'll probably have to take a connecting flight from Kolkata, which will take you to the one of the cities in the Siberian district with a hault at 1 or 2 stops.
Almost all major cities have airports, which connect Siberia to other parts of Russia as well as the rest of the world.
The Trans-Siberian Railway enables you to hop into majority of the towns and cities in the district.
Some irregular ferries are available between Vladivostok (Russia) and Niigata and Fushiki (Japanese ports).
Public transportation in cities of Siberia region varies- some cities like Novosibirsk have metro, buses and trolley buses; while small towns might or might not have anything more than a mini bus or taxi. It is recommended to work out on the trip plans clearly, and then doing some research as per the cities you plan to visit. Travelling from one city to other is made convenient by railway, though you can also take a flight.