By bus
JETT buses, both ordinary and all-inclusive guided tour, connect to Amman and Aqaba. Other tourists come with organized groups, including daily trips from Eilatm. Tours to Petra from Taba, Sinai and Sharm el Sheikh are also gaining popularity with charter tourism.
By minibus
The minibus from Wadi Rum costs 3 Jordanian Dinars (JD) each. It takes 1.5 hours to get to Petra. Have the Rum Guesthouse or your tour operator call the bus owner the day before to arrange an exact time for pick up. The bus usually leaves from Wadi Rum at 8:30 in the morning, but may be delayed due to weather or tour groups coming the other way.
There are also minibuses from Amman departing from the Wadabat bus station - these leave when full, and tourists are almost always charged 3 JD to get on. DO NOT allow the drivers to charge you for your luggage, as they might sometimes try to do, considering you're already paying more than the locals (who pay 2-2.5JD).
By taxi
Taxi is also a viable option. For 75 JD or less (depending on how much you haggle) you may be able to get a private taxi from Amman to Petra and back, including the driver waiting around for 6 hours. It would cost 24 JD for two persons to travel by JETT bus, and allow you to see almost the entire site in an (exhausting) day trip.
If you get there renting a minibus with a driver in the hotel at the Dead Sea, the one-way price would be 140 JD.