Before going to Amman, it is advisable to keep in mind that even though it is a capital city it is not so much organized as other European cities. This is the result of the fast growth of the city sine it was named capital. That is why you might experience many inconsistencies in names of landmarks and directions you are given. That is why it is important that before you go Amman you learn the locations of the 8 circles of Zahran Street; the district of Amman and the hills. These will help you get around easier and it will help people to point you the right way if you ask for directions.
Once in Amman there are several ways that you can get around the city.
By a rental car
There are several companies for renting cars in Jordan. Some of the companies might even give you a free driver if you rent a car from them. If you decide to rent a car then make sure that you get an extra liability insurance for your car. If you decide to drive in the city by yourself keep in mind that many streets do not have lane markings, the locals do not really respect the actuals laws and drive according to some conventional rules. It is best if you drive slow and on the right side. The local drivers and not really aggressive, but make sure that you have your eyes wide open when driving.
The taxis in Amman are colored yellow and grey. They are really easy to find anywhere in the city and you can hail them in the street. The yellow taxis have a logo on the doors, whereas the grey ones have an advertisement on the top. It is best that you hail these kinds of taxis because they are legal taxis from the city. There are also white taxis which are shared taxis and they have a specific route. Taxis are required to use the meters. If someone offers you a free "welcome to Jordan" ride do not accept because you will end up paying a lot more than you should have paid.
Amman has big, municipal buses that go around the city. Make sure that you pay the exact fare for the bus because the drivers do not have change. You can also get a bus fare cash card which you can fill with money and swipe it past a reader as you enter the bus. Most of the buses are numbered so you could learn the numbers of the buses that will take you to the places that you want to go to.
Amman isn't exactly designed for pedestrians. It is in fact very opposite of that, with big hills, no pavements, no crossings and streets like labyrinths. But, walking through the city is the best way to make sure that you have seen every corner of it and to connect with the local life there. If you decide to walk make sure that you get a decent map or you could ask the locals for directions. They are always willing to help. Just make sure that you keep an eye on the traffic and keep in mind that you will not get to your destination in the preferred time.