You will have to change a couple of flights to get to Yellowstone from India. First a flight to the US, then a flight to one of the airports near the national park will have to be taken. The flights to the airports nearby are mostly provided by Delta and United Airlines only. You will have many different options of approaching Yellowstone depending on your itinerary.
You can apply for a tourist visa for the United States at your nearest US Embassy. Basic requirements include proof that you are staying for only a limited amount of time, proof that you can financially support yourself during your stay and also proof that you have social or economic ties in your country that will ensure your return. You can confirm all the other documents required at the US Embassy's website.
The airports at Jackson Hole and West Yellowstone are the prime airports connecting Yellowstone. The latter is only commercially operating during the peak season only (June to September), and Jackson Hole has flights from Salt Lake City and Denver throughout the year, and from a few other locations during the on season. The Yellowstone Regional Airport in Cody also has year round service provided by Delta.
Getting to Yellowstone via car is the most popular way. Car rental services are available at all the gateway towns, and the park is best accessed by having your own car as well. There are 5 entry points to the park, Gardiner, Montana (north), Silver Gate and Crooke City (northeast), Cody, Wyoming (east), Jackson Hole (south), and West Yellowstone (west).
Rail connections directly to Yellowstone do not exist. Amtrak provides trains to Salt Lake City from where you can catch a bus to Jackson, one of the 5 gateway towns of Yellowstone.
Getting around Yellowstone is best done via car or tour bus. There is no established public transport system inside the park. Cycling is also a popular option to explore Yellowstone. But that being said, to cover the expanse of the park, it is best to rent a car.