The major international airports in India, which serve traffic from all over the world, are in Delhi, Mumbai, Kolkata, Chennai, Bangalore and Hyderabad. Airports tend to be on the outskirts of cities. Pre-paid taxi services and auto-rickshaws are stationed outside the terminus to get you into the city.
Railways do not cut across international borders in this part of the world except the Samjhauta (i.e. "understanding"!) Express that runs between Amritsar (India) and Lahore (Pakistan). But since the running of the train is subject to relations between the two countries it would be prudent to check if the train is running. However, in many cases it is possible to travel till the border by train.
Except the Lahore-Delhi bus which used to run 4 times a week, but is now erratic, there are no cross border coach services. It is possible to drive into India with the requisite paperwork in order. Even driving in from Nepal now requires a permit. Permits may be arranged through the Indian embassy in your country.