The city centre in Amman is actually quite small. You can walk around it in less than a day and see all of the sights by foot. The Turkish Mosque is a must-see, beautifully decorated, you cannot actually go inside but just gawp at its stunning design from the outskirts.
The Citadel sits on a hill above the city and is the perfect place for panoramic views of Amman. It's peaceful, historic and the best place to people watch. The museum has artifacts dating back centuries and will take you back through the years, teaching you all about the Islamic and Jordanian culture.
I travelled to Jordan as a solo female. If you're looking to travel alone to Jordan, there's lots of advice to help you plan your trip on the Girl about the Globe website.