We travelled for sixteen days as backpackers around Jordan in October just gone (2011).
You can do Jordan reasonably cheaply as we did, because there is a lot of long distance and local transport which is very cheap by European standards. Except for our long journey from Madaba to Petra (which is not well served by point to point public transport) and our two day trips to the Desert Palaces east of Amman, and down to the Baptism site at Bethany (I thought the entrance fee for the latter was a rip-off for what we were allowed to see and the time allowed), we did the rest of our journeys by servees (shared taxi), minibus or bus, rather than by hired car. Public transport is easy, safe and very cheap. And you are meeting real Jordanians all the time.
We spent three days seeing Petra (we bought a multi-day ticket) but even those weren't enough for us: we loved the place and would go back on another trip. Don't miss it, it's one of the great sites/sights of the world. Jerash is great and we loved the Desert Palaces out in the desert east of Amman (we paid 18JOD each for the day trip to see them - booked through one of the Amman hotels .. you have to book and pay the day before you want to go). We also visited the wonderful town of Madaba (stayed there twice because we liked the ambience so much), Umm Qais, Irbid and Amman.
You can eat in local falafel or shawarma places (really cheap) or find other local cheap diners. There are always street vendors in the town centres and at bus stations selling cheap fruit and nibbles at a great price.
Just read and digest the guide books and, if in doubt about something being offered, check it out with someone in authority - on-duty policeman, bus station supervisor, an older shop owner - or your hotel manager. They are always willing to help because the country depends so much on tourism that they want to maintain its good reputation abroad.