2 days in Petra might suffice to get a real good look at the ruins
Whether or not Amman is "worthwhile visiting" is purely a matter of personal opinion. My opinion is that it is not, others have differing opinions.
As long as you're going to Petra I would recommend spending a day and night in Wadi Rum, staying in a Bedouin camp.
Good luck.
For public transport in Israel, see here - an "umbrella" site covering all bus companies as well as trains.
Generally speaking, the Galilee and Golan are sparsely served by public transport, so renting a car definitely makes sense there. You don't want or need a car in Tel Aviv or Jerusalem, where it will be more of a liability than anything because parking is almost impossible to find, as has already been said.
Also note that there is no public transport on Shabbat - the Sabbath, from Friday late afternoon until after dark on Saturday, so again a car will be useful then. Bear in mind that except at the airport, car rental offices close at around 1 p.m. on Fridays and reopen Sunday morning. One exception to the no Shabbat transport is the sherut (shared taxi) between Tel Aviv and Jerusalem that runs 24/7. Another exception is the bus from Tel Aviv to Eilat and reverse, which runs on Saturdays starting around mid-morning. In east (Arab) Jerusalem and the Palestinian Authority areas, public transport runs as normal throughout the week, so if you are planning on going to Bethlehem, for example, Saturday is a good day to go. Buses to Bethlehem leave from outside the Damascus Gate of Jerusalem's Old City.

Some people might beg to differ but I would say Amman is not worth a visit.
I suspect Dude set foot in Petra only with one of these giant tourist buses that take you to have a glimpse of the site on a one-day trip from Eilat - a classic tourist trap. You need 2.5 days on location to see the entire site. In one day you will have only a very rough idea of the site and will be frustrated. In two days you will have seen most of the highlights. The Lonely Planet guidebook explains very well the different paths inside the site.

A one day trip from Eilat is not a "classic tourist trap". Catw is correct (for once) that you need 2.5 days to see the entire site, but many people do not have the time, stamina or interest to spend so much time there. A one day tour will allow you to walk the length of the canyon and see all the highlights without being rushed. Catw's statement is equivalent to saying "You need 2.5 days to see the Louvre so any tour that takes you around for 3 hours is a classic tourist trap". Every tourist faces the perennial problem of allotting limited time to unlimited attractions, i.e. an extra day at Petra could mean not seeing something else of interest or importance, and sometimes one day is the only solution.
Regarding Amman, it's a nice place but as a tourist I wouldn't go out of my way to visit there, except perhaps as a base for visiting Madaba, Jerash and Mt. Nebo.

A one day trip to Petra from Eilat is a classic tourist trap. It does not make any sense. It's as ridiculous as the one-day trips to Jerusalem from Taba. You do not have time to see anything. You need to be extremely naive not to realize it.
As I wrote above, if you're on a tight schedule you can do with one full day in Petra (and two nights). However you will be frustrated if you spend less than two full days. You need 2.5 days to really see the site.
Petra is one of the most magnificent places on earth and it's huge. Entrance to the site is also very expensive, so better enjoy it for two days (the price drops if you spend more than one day).

Catw: everything you say is correct but some tourists (a) don't have the time (b) don't have the strength or stamina (c) aren't particularly interested in ruins. You have made your point and now please give the tourists credit for sufficient intelligence to make their decision.

Actually Catw, I spent 2 days there, which is al most people have ad I traveled there on a shared taxi from Aqaba. I entered on both of those days and walked though out most of the main sites. SO careful with you assumptions.
Catw
Just a quick question. What if I did come in on a tour bus, whic I didn't, what of it? Shoud I be seen as a lesser person for it? For someone who claims ot be so well traveled. you sure come of as being ignorant.
Edited by: DUDE057

Personally I love Amman, but there is not much there for a tourist, then again you may find it interesting to simply 'be' in a thriving Arab city. One thought is that you go from Eilat up to Petra, then go on to Amman, have one night and return over King Hussein/Allenby Bridge? If you are heading back to the centre its a similar time scale to going back to Eilat, staying in Eilat and then travelling up - but it depends on what interests you.
G