Pompeii is an easy reach via car or train. It, however, doesn't have an airport and you will have to traverse via Naples, Sorrento or Rome.
Flights from India to Italy are available from nearly all the international airports of the country. Airlines such as Qatar, Etihad, Air India, Jet Airways and many other airlines provide flights to Rome. Air India operates direct flights only from New Delhi and Bombay. From Rome, you can take a train to Pompeii, or rent a car, both of which will take you approximately 2 hours.
Italy is part of the Schengen Area of Europe. You can apply for a Schengen Visa at your nearest Italian embassy. If you already have a valid Schengen visa, it is very much usable. The Schengen visa can be used for 3 months within a 6 month validity period. All information related to the exact requirements of the application easily be enquired from the embassy. You can apply and get the details at http://www.vfs-italy.co.in/
Pompeii does not have an airport. A flight has to be taken to Naples or Rome, and then a train or bus to reach Pompeii.
You can hire a car and drive down from neighbouring cities, but you will have to deal with the heavy traffic near the modern city of Pompeii. Parking can also be a hassle at times.
The rail would be the best way to reach Pompeii. You can get on the Circumvesuviana Napoli-Sorrento line from either Naples or Sorrento and get off at Pompeii Scavi. If you're coming from Rome, you can get on at the Termini station, the tickets costs around EUR 11. You can also get on a local train from Salerno. You can keep your bags at the station for a small fee. There is a Tourist Information Office close to the Pompeii Scavi station.
Pompeii is basically a walking site. However, you can also rent a bicycle. The surface is rather uneven, so most tourists prefer walking. Wearing a comfortable pair of shoes and carrying a lot of water is a good idea. One day tickets are available for EUR 13 and a five site pass costs EUR 20 which also includes entry to Herculaneum.