Tons of things to do in Amman:
During the day you can see:
1. Roman Ruins (the Citadel, Amphitheater, ancient Basilica, etc.)
2. Archaeological museums
3. Royal Automobile Museum
4. Shop at Souk Jara or Sweifeih Market
5. If you are adventurous, you can take a 30-40 minute car ride to the Dead Sea and/or Baptism Site at the Jordan River
Food:
1. Hashem (old city) - for typical breakfast and/or late night post-drinking meal of hummus, salads, falafels, french fries, etc. Old school but lots of fun
2. Fakhr al-Din (First Circle) - probably the best kabobs in town
3. Reem Shawarma (Second Circle)
Entertainment
1. Tons of bars/pubs in the city
2. Rainbow Street for people watching - this is where the young adults congregate at night
3. You can catch an excellent meal/drinks/hookah at any of the cafes or restaurants in town
PM me with more specific questions. Enjoy Jordan!