Bihar the most ancient place in Eastern India and first to gain what can be called civilisation, culminating in Magadh Empire.
In some regions Hindi is used. In the cities you will also find people knowing English. Most of the states have their own regional language though. Bengali is spoken in West Bengal, Odia (also spelled Oriya) in Odisha and Nepali is used in Sikkim.
The best way to address anyone in Eastern India especially in the Bengal region is 'Namashkar' - loosely translated to 'Hello' and don't forgot to thank - 'Dhanyabad'
There is an international airport in Gaya and Kolkata.
Trains connect the cities and there should be no problem reaching Kolkata as well as other cities from Delhi or any other part of India.
From Bangladesh, there are a number of land entry points to India. The most common way is the regular air-conditioned, and comfortable bus services from Dhaka to Calcutta/Kolkata via Haridispur (India)/Benapole (Bangladesh) border post. Bus companies 'Shohag', 'Green Line', 'Shyamoli' and others operate daily bus services under the label of the state owned West Bengal Surface Transport Service Corporation (WBSTSC) and the Bangladesh Road Transport Corporation (BRTC). WBSTSC operates buses from Calcutta every Monday, Wednesday, and Friday while from Dhaka the days leave on Tuesday, Thursday, and Saturday. BRTC also operates the buses from Calcutta and Dhaka on the same days. The normal journey time taken by these busses is around 12 hours with a oneway fare of Rs. 400-450 or BDT(Taka)600-800, roughly USD 8-10.
Another daily bus service by 'Shyamoli' and others under the BRTC label from Dhaka connects Siliguri, but the busses in this route do not cross the Changrabanda/Burimari or Burungamari border post. Rather, passengers reaching the border have to clear customs, walk a few hundred yards to cross the border and board the awaiting connecting busses on the other end for the final destination. Ticket for Dhaka-Siliguri-Dhaka route costs BDT 1600, roughly USD 20-25 depending on conversion rates. Tickets are purchased either in Dhaka or in Siliguri.
Other entry points from Bangladesh are Hili, Chilahati/Haldibari, Banglaband border posts for entry to West Bengal.
Local trains are a good way to get from one point to another. Although sometimes overcrowded it sure is an experience to see the locals getting around and buy some snacks from the passing sellers.
It is also possible to take a bus to get to other cities.
Kolkata has several places to see, do not miss the mighty Hooghli (the eastern distributary of Ganges passing through West Bengal) and the well maintained remains of the colonial British Raj such as Victoria Memorial. Mother Theresa's work is a lot more visible in Kolkata than in any other Indian city and access to the Missionaries of Charity is generally open to public. Some intense Hindu places of worship include Kalighat and Dakshineshwar.