Lucerne is an easy reach via car or train. It, however, doesn't have an airport and you will have to land at Zurich or Berne and then take a train.
Flights from India to Switerland are available from nearly all the international airports of the country. Airlines such as Qatar, Etihad, Air India, Jet Airways and many pther airlines provide flights to Zurich. Swiss Air and Air India operate direct flights only from New Delhi and Bombay. From Zurich, you can take a train to Lucerne, or rent a car, both of which will take you approximately another hour.
Switzerland is part of the Schengen Area of Europe. You can apply for a Schengen Visa at your nearest Swiss 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 can easily be enquired from the embassy.
Lucerne does not have an airport. A flight has to be taken to Zurich or Berne, and then a train to reach Lucerne.
Driving to Lucerne is going to make you witness some of the most stunning and picturesque views Switzerland has to offer. The roads are magnificent too, adding to the entire spectacle of driving down. Lucerne is at a distance of about 45 minutes from Zurich, and an hour from Berne and Basel. Renting cars in Zurich is fairly simple. You'll find a lot of options at the airport itself.
The train system in Switzerland is actually quite sophistciated. It covers almost the entire country and the views and sceneries are just breathtaking. Trains from Zurich will take you about 50 minutes to an hour, and an hour and 15 - 20 minutes from Basel and Berne.
Buses and local train networks are ideal forms of commuting in the city. The entire region is divided into zones and Lucerne City is Zone 10. A day pass for zone 10 will cost CHF 10. The pass will allow you to cover all the attractions of the city. Purchasing of tickets can be done through cash, credit card, or through a phone app called OV ticket. A fine of CHF 90 can be levied if you're found travelling without a ticket or with an invalid ticket.