Most visitors reach Petra from Amman — about three hours via the Desert Highway, or four to five if you take the slower King's Highway through Madaba, Karak, and Shobak (which is the better drive). The JETT bus runs once a day from Amman to Wadi Musa at 6:30 am, returning at 5 pm; the trip takes about four hours and is the simplest option without a car.
From Aqaba on the Red Sea it is about two hours up to Wadi Musa, an easy day either way. From Wadi Rum the drive is about ninety minutes; a number of Bedouin operators run combined trips. Shared service taxis depart from Aqaba and Wadi Rum in the morning if there are enough passengers.
For arrivals from Israel/Palestine, the King Hussein Bridge (north of Jerusalem) is the only crossing that does not require a Jordanian visa in advance for many nationalities. From there it is a long taxi or hire-car day down to Petra.