I did this a few years ago, but I'm a guy.
What do you intend to see/do, and how many days do you have? Are you planning to do the crossing from Amman or near Petra?
Must sees would indeed be Petra. I would recommend at least 2 days for Petra, although some people rush through it in 1 afternoon. I took an early morning bus from Amman to Petra, then back to Amman 2 days later. From Amman, I took a taxi to the border crossing, took the special bus to the Israeli side, and then finally another bus to Jerusalem.
Other must sees are the Dead Sea and of course Jerusalem. Other places of interest to Christians would be Bethlehem, Nazareth, and River Jordan. Amman is a jumpoff point but itself doesn't have much in way of tourist attractions.
If crossing from Amman, you'll be using the King Hussein Bridge (called the Allenby Bridge in Israel). At the Israeli immigration, you'll get your passport stamped. You can request for your passport not to be stamped, and they'll instead stamp a piece of paper attached to your passport. Some people do this because some conservative Islamic countries do not allow entry to tourists with an Israeli stamp in their passports.
I didn't do this though, because I also heard this request may piss off the Israeli immigration guy and he may refuse you entry into Israel.
Bring some US dollars to change into israeli shekels at the border as there were no ATMs when I was there. Then grab a shuttle bus from the border into Jerusalem.
Jerusalem is divided into the largely Jewish West and the Muslim East. The West is much more developed and modern. The East is more rundown. The difference in the level of development between the 2 sides is quite striking.
I didn't know this when I went, so I unknowingly booked a hotel in the east, on Mt Olive. So each night I had to walk past groups of disillusioned or disenchanted youths. Once a group of Palestinian youths asked me where I was from, and then uttered "I kill you". Now he was likely just pranking me, but as the only foreigner in a largely Palestinian environment, that sounded quite uncomfortable. If I could do it again, I would probably stay in Western Jerusalem.
Has anyone been to Jordan / Israel?
I'm planning a solo (female) trip to Jordan then to Israel (overland) over the two public holidays in May. If time permits, I'm thinking of going to Dahab, Egypt to do some diving.
Would appreciate and tips and advice (like the must sees besides Petra etc).
Thanks!