Go up to the Citidel take some wonderful photos of the city to send back home to get everyone excited. Try some shopping in downtown, Jabal Amman, etc. Although it's crazy now because of Eid it's part of the experience. Go to a cafe and have some shisha, Turkish coffee, and knafa. An afternoon at the Dead Sea is one of my personal favorite things. There are plenty of things to do and visit depending n your tastes. Of course the big things like Petra you'll probably want to save for when your family comes.
As far as preparing for the move I can tell you the things I regret not bringing with me. If you think you will need any portable electronics (laptops, iPads, cell phones) in the near future buy them before you come as stuff like that costs twice as much here. Also kitchen stuff (pots, silver wear, knives) other than dishes and glasses. Of course unfortunately, most of your appliances won't work here. The converters don't work (220 to 110) they just end in a puff of smoke. If you wear men's shoes size 11 or above or wide feet you'll have a hard time finding ones that fit here. Shoes don't come in half sizes either and I've yet to find a pair that fit me here other than at the "American outlet" stores. Also to note that it gets cold here in the winter. My apartment doesn't have heat other than a portable gas heater we use which is the norm here. I'm from a very cold and snowy part of the USA and I thought I'd be able to handle it but not having central heat is a whole different animal, 50° in your house when you just got out of the shower seems a lot colder than 50° outside. Bring lots of long underwear and warm slippers. Especially if you are coming in the winter!