There are always so many questions how to get to Petra and Wadi Rum. I went to Eilat and found some people to travel with. Together we took a cab to the border with Jordan - 5 minutes and ooh I forget how many shekels maybe 20 (5$) then you walk across the Israeli border then the Jordanian border. As an American you don't need to have a visa in advance and you get one free (although I paid $37 for min and fedex!!). Once on the other side taxis bargain with you. We paid 35 Jordanian - like 50 to be taken right to Wadi Rum. We learned we should have paid 25. Many people take a taxi to Aqaba - around 10 Jordanian (1.5 to dollar) then a bus to Wadi Rum - bout an hour. Petra is 3 hours from Aqaba. There is one local bus a day, maybe 2 that runs from Wadi Rum to Petra.
My friends needed to get from Petra back to Aqaba same day and while the taxi was supposed to be 50 or so Jordanian, they got a deal for 1/2 the price to the border.....the 8:30 am bus gets in like 11:00 or 11:30.