There are many options to travel between Ataturk Airport and City Center.
Aksaray, Emniyet - Fatih, Topkapi - Ulubatli, Bayrampasa - Maltepe, Sagmalcilar, Kartaltepe - Kocatepe, Esenler Otogar, Terazidere, Davutpasa - YTU, Merter, Zeytinburnu, Bakirkoy - Incirli, Bahcelievler, Atakoy - Sirinevler, Yenibosna, DTM Istanbul Fuar Merkezi, Ataturk Airport
Using Airport Metro can be good option when you travel light. You can easily go to Old City area by using Airport Metro and Tram. There is a connection between Airport Metro and Tram at Zeytinburnu Station, you need to change to Tram at this station. By doing this you can travel to main touristic areas of Istanbul. You can find the map for the Metro and Tram here
New fares for the lines M1 (Aksaray-Ataturk Airport), M2 (Sishane-Haciosman), T1 (Bagcilar-Kabatas), T3 (Kadikoy-Moda), T4 (Topkapi-Habibler), F1 (Taksim –Kabatas), and Macka-Taskisla and Eyup-Pierre Loti Aerial Cable Car
Full Fare for Electronic Ticket (Akbil – IstanbulCard):
Student Fare for Electronic Ticket (Akbil – IstanbulCard):
Discounted Fare for Electronic Ticket (Akbil – IstanbulCard):
You can find more detailed information here
This option is good when you are travelling light and your destination should be old city. Otherwise you will need to use other transportation alternatives.
There is a shuttle company called Havatas which have shuttle buses every 30 minutes from/to Airports. They are fully reliable and comfortable. You can fit many luggage to this buses. But they have services to Taksim and Yenikapi IDO Seaport. Please keep in mind may need to walk to your hotel after Havatas, this can be very tiring with luggages. Fares for this option are 10 TL for Taksim and 5 TL for Yenikapi IDO.
You can find the timetables and more detailed information here
Using taxi is a good option if your distance is not far. A taxi will cost around 45 TL from airport to Old City and 50 TL for Taksim. Aggreing on a price with the driver can be good, because they charge by the distance. Please read this article
to have an idea what kind of problems you may face with the taxi drivers.
This is the best option you can take for transportation from/to Airports in Istanbul. In Istanbul there are many transfer companies and they have good prices, most of them are good companies and fully reliable. You can pre-book your vehicle by internet, driver meets you in the airport on your arrival or at your hotel on your departure. They have fixed prices and no-extra charges.