When it comes to getting around the city of Dubai, the metro is a great way to go. The Metro connects you to many of the largest malls, the old town, and many attractions. You can visit the Burj Khalifa, go out to the end of the Palm, and even visit the seaside at the Dubai Marina either right at or very near the Metro. One of the key places that isn't on the Metro system is the area near the Burj Al Arab, including Wild Wati water park and Jurmera beach.
There are many different types of tickets available for the system, for tourists the easiest way to go is with a day pass. This pass will let you travel as many times as you want on the Metro in all zones, as well as on the bus, for just 14.00 dirham. The main benefit for the all day ticket is you don't have to worry about the zones, which can be fairly complicated and limiting.
The lines themselves are fairly easy to figure out since there are really only two lines. One line travels in a straight line from one end of the city to the other, while the second line does a U around the creek. There are a few offshoots, in particular the light rail at the Dubai Marina, and the monorail at the Palm as well.
It is important to make sure you enter the proper car when boarding the train. One car will always be reserved for first class tickets, another car is dedicated for women only. The women only car times change, however if you're a male and unfamiliar with the system it's simplest to avoid this car all of the time.