Coach services from both İstanbul Atatürk Airport and Sabiha Gökçen Airport to Taksim Area are given by Havabus company. You can take a coach from both the domestic and international areas of both airports. A one way ride between Taksim and Atatürk Airport is 12 Turkish Liras. A one way ride between Taksim and Sabiha Gökçen Airport is 15 Turkish Liras. Service hours and more information can be found at Havataş's website: http://www.havabus.com/default.aspx
Taxis are easy to spot as they are all bright yellow in colour. All have a taxi meter which are switched on at the beginning of your journey. If you like, the taxi drivers may inform you before the journey about the approximate cost you are going to pay with respect to your distance.
Approximate cost from Istanbul Ataturk Airport to Taksim is about 20 EURO, and from Sabiha Gökçen Airport to Taksim is about 30 EURO. Please note that these are estimated on daily traffic with one route. The actual costs may vary.