Istanbul has 2 airports: Sabiha Gökçen Istanbul Airport and Atatürk. Sabiha Gökçen is located on the Asian side of Istanbul.
Metro - Ataturk Airport to the centre
Jump on the metro to Zeytinburnu. Once there transfer onto the Zeytinburnu-Kabatas tram which will take you to Sultanahnet Square, Sirkeci Station or Galata Bridge. You can also travel to the ferry docks or to Istanbul’s main bus station Büyük Otogar.
From both airports you can get the Havatas shuttle bus to the centre. From Atatürk it will take around an hour to Taksim, because of the traffic. A ticket costs around 10 TL. From Sabiha Gökçen it will take around half an hour to get to Kadikoy and you'll pay around 13TL .
From Atatürk it's easy to get a taxi straight to your apartment. A taxi to Sultanahmet should cost you around 45-50 TL. From Sabiha Gökçen it is less convenient to get a taxi. It costs around 80TL to get into the centre and it takes more than an hour.