Note: This post was written in 2014.
Amman is a massive, sprawling city. We stayed in the older part. Older, as in, going back to the time where it was part of the Roman Empire, and even earlier. People are quite conservative in this area. The only people we saw wearing shirts that didn’t cover them to the wrist were tourists. Couples should refrain from public displays of affection, including holding hands, as this is not appropriate. Two men, however, may hold hands in public, as it is merely a sign of friendship.
We walked up one of the many big hills behind our hotel and explored the Citadel. It was well worth the 2 JOD/$2.82 entry fee. This place is ancient, possibly being inhabited since the Neolithic period.
We also explored the Roman Theater. Entry fee was 2 JOD/$2.82. There are two museums on the site, one on the left, the other on the right as you enter the main area of the amphitheater, make sure not to overlook them. The climb up the steps of the theater was quite the workout, and to be honest, a little scary. Some of the steps have been worn down quite a bit. I saw women in high heels climbing the steps; how they managed to not break their necks I have no idea. (The steps on the far left as you climb up appear to have been replaced and are a little safer to climb up/down than the ones in the center.)
For some reason neither of us took pictures of the amphitheater while we were in it. I blame the cold windy weather that day.