The best way to go from India would be to take a flight from your desired Indian city to Amman, and then take a cab or a bus from there and head straight to Petra.
Queen Alia International Airport at Zizya, 30 kilometres from Jordanês capital Amman, and King Hussein International Airport at Aqaba are the two airports in Jordan. Regular flights are available from international airports such as New Delhi, Chennai, Bangalore, Mumbai, Pune, Kochi, Kolkata, Jaipur to Amman and Aqaba as well. Air Arabia, Oman Air, Gulf Air, Kuwait Airways, Jet Airways, Etihad Airways, Emirates, Turkish Airlines, Lufthansa, Air Arabia, Royal Jordanian Airlines, Air India, Egypt Air fly from India to Jordan. If coming from Israel, flights from India mostly land at Tel Aviv Yafo in Tel Aviv. Connecting flights are available to Eilat.
Taxis are available from both cities to Petra. The white painted service taxis ply on fixed routes and can be shared. Private taxis are painted yellow and can be hailed on the street or near hotels. Car rentals are available and details are provided on the Jordanian Embassy website. If coming from Israel, the best option is to cross at the South Israeli Arava Border near Eilat. It takes about one hour twenty minutes to get from Eilat airport in Israel to the border (including the crossing of the border) and two hours to drive from the border to Petra. Visa to Jordan is not given at the borders and have to be obtained prior to reaching the border. Border taxes have to be paid on crossing the border. One can enter from Syria and Egypt by road as well.
Air-conditioned JETT (Jordan Express Tourist Transportation Company) buses are available daily from Amman to Petra in the morning and they return in the evening. It takes about 3 hours each way. Online reservations can be made in advance at their website http://www.jett.com.jo/Default.aspx. In addition, public buses are available from Mujamaa Janobi Station to Wadi Musa daily from 09.00 to 13.00 hours and return between 06.00 hours to 13.00hours to Amman. Public buses are available from Aqaba to Petra from the bus station next to Aqaba Police Station. These take about two hours. Though there are no fixed timings, buses are available from 06.00 hours onwards.