Visit the Theater in downtown Amman. It's about 5JD for a guided tour. It's not necessary, but it was nice. The guide was an old fellow, with very few teeth, but very nice and very knowledgeable.
Fondest memory: Wandering around the small market that is about 200 meters from the Roman ruins. Several good food stalls there...
Visit the ancient Amphitheatre in the city centre.
Fondest memory: Sitting in the theatre watching the people and enjoying the architecture.