score:3
Indeed there isn't that much to see in Amman, not a very touristic city.
There wasn't much recreational part to Jordanian side of Dead Sea, there is only one resort and the only place to stay there are quite expensive hotels (Marriott and MΓΆvenpick, both with room prices $200+ per night), so do it as a day trip or on your way down south. But I'd say it was quite a let down. If you have an option to see Dead Sea from the Israeli side, you'd be much better of doing that.
As Grzenio suggest, Jerash is close by to Amman, and definitively worth a visit.
From there I'd continue down south, if you haven't seen Krak des Chevaliers in Syria, then castle of Al Karak is second best crusader castle to see.
I'd suggest that you arrive to hotel in Wadi Musa evening before, as you're gonna need full day to see all of Petra. This is as you already know the thing to see in Jordan. (see also Getting to Petra from Amman in Jordan)
Further down south, just short of reaching Aqaba, you can visit Wadi Rum. It just a desert and rocks, but still it is quite spectacular. You can make Wadi Rum a day-trip from Wadi Musa, or go there on your way further down south.
And of course Aquaba, there you can have lot of fun snorkeling in the reefs. City itself is also quite touristic, so you have lot of places to eat out etc.
As for transport, there are regular coach services between main cities. They are cheap and quite comfy, air-conditioned. Reaching touristic destinations might require you to take less regular mini-bus services. Really cheap, but comfort may vary. Other option is just take a taxi, they aren't very cheap, but still quite affordable method of transport. Especially if you on a tight schedule.
Upvote:2
You already received some good suggestions but i'd like to add my personal experience.
If you absolutely want to have your base in Amman here is what you can do:
You can also evaluate this variation. Go to Petra on day 3 and spend a night in the super nice Valentine's Inn where you can meet fellow travelers and organize with them a trip on day 4 in the Wadi Rum, absolutely amazing. You can hire and share a local guide to get there.
If you don't necessarily need to stay in Ammanm I would suggest you to head down to Petra the first day, get in love with it and stay at least 2 days (I stayed for three days). Then you can go to the Wadi Rum, hire a local guide as already suggested, and spend a night in the desert too. Aqaba isn't really interesting but from Petra you can reach it quite easily or you can head up north to Amman and spend a day there before leaving (or skip it entirely and have fun in Dead Sea).
Upvote:4
Amman is not so interesting, so I would suggest that a day is more than enough if you have that little time. A very good and easy day trip from Amman is Jerash - easy to get to by bus. Aquaba is also easy to get to, because there are many fast buses there from Amman (3-4h, probably every hour or so).
Petra on the other hand might be much more difficult to get to for a day trip (its THE thing to see in Jordan though, so you must find a way :) - we ended up taking a taxi from Amman (it wasn't cheap 6 years ago). Probably you will need to spend at least one night there. Also its quite big, so a full day is needed. You might try to find some organized day trips, but I can't really help you here.