The best way to see Petra is either on an organized tour, which takes all the hassle out of it, or DIY which is more complicated. A British company named Voyages Jules Verne (VJV.com) are very professional and competitive,taking in all the sites, including Petra, Wadi Rum, Dead Sea, Madaba,Karak, Mount Nebo, Jerash and the Desert Castles to the east. They fly into Aqaba, where you stay for a few days at the end of the tour to relax by the Red Sea. They have chartered their own flight. Otherwise I think you will pay more just for the flight. You can fly into Amman (where we also stayed) with Royal Jordanian, but it will be more hassle and expensive to do it this way. I have been twice and love the history and landscapes and people.
I spoke to someone earlier on a different travel site and it did seem easier and cheaper to use VJV.
Good luck with what ever you decide.
Daisymay