Taxis are good transportation to get around Istanbul.  They are inexpensive, have meters and most of the time easy to find.  Your hotel can call one for you as well. After midnight, taxi drivers charge double.

Public  transportation works well, is inexpensive, but at least the tram between Zeytinburnu and Kabatas is often quite crowded. There are 2 metro lines. One from the Atatürk Ariport to Aksaray and one from 4 Levent to Taksim. There are 4 tram lines and the most useful one for turists is the one between Zeytinburnu (where you can change for the metro to Atatürk Airport) and Kabatas (near Dolmabahce Palace). It passes many major attractions in the historic center of Old Istanbul like the Blue Mosque, Hagia Sofia and the Topkapi Palace. It passes the bridge over the Golden Horn between the Old and the New Istanbul. From Kabatas there is a new funicular up to Taksim. From Taksim you can take the nostagic tram through the major shopping street of Pera (Beyoglu) and then the old funicular (Tünel) down to the bridge over the Golden Horn. Jetons for single trips are sold right at the stations and costs 2 YTL (as of 15 Aug 2011).

See  transportation map at:  http://www.istanbul-ulasim.com.tr/en/