Start your day downtown at the Citadel. Walk amongst the ruins, specifically the Temple of Heracles and the Umayyad Palace. Check out the small but informative National Archaeology Museum to get some context. The views from the Citadel are spectacular. On a clear day, you can see the surrounding hills for miles, each covered in Amman's signature white-cube apartment blocks.
Next, head down the hill and across the street to the Roman Amphitheater. Marvel at one of the Middle East's most spectacular Roman ruins before heading the few blocks to downtown (or Wasit Al Balad). Amman is a seriously decentralised city, but Wasit Al Balad is its beating heart. Spend a few hours exploring all the different souqs and side alleys. Get some lunch at one of the many vendors and spend some time people watching from a balcony cafe. This is also a great place to buy some souvenirs.
Go across town to the University of Jordan. There aren't many parks in the city, and the tree covered campus is a good alternative. Enjoy the collegial atmosphere and walk along the tree-lined paths. The campus features a nice little museum about the history and archaeology of Jordan.
In the evening check out Rainbow Street in Jebel Amman. This is the trendiest spot in the city. If you want to enjoy the nightlife, hit up one of the bars or clubs. If not, enjoy some coffee at a cafe. This is a great place to get dinner too. Enjoy some of the city's signature shawarma.