Istanbul has two Airports. The main Airport is located on the European side of Istanbul and called as Ataturk Airport (IST), The airport was modernised on 2000 and again extended in 2004. The airport can host up to 20 million Passengers a year.
When you arrive to Ataturk Airport your first stop will be the Passport control. If you need to obtain a visa to Enter to turkey, you can get it at the office just next to the passport control. Be sure to get your visa before getting in the Passport Control line. After the passport control, there are 11 baggage bands. Four on left side and seven on right side. There is a digital TV which shows baggage bands according to the flights just after the pass port control. So you shall know where to go, by following the Screen.
After the baggage claim, there is the customs. Turkey uses a Red/Green channel system. Unless you are spot checked you can pass the custom and exit from the arriving hall. Just after the exit, you will be on the welcome hall where you can find Exchange offices, banks, information desk and cafes.. HAVAS buses take you from Ataturk Airport to Taksim. Here is the schedule: http://www.havas.net/en/shuttle-parki.... This costs 10 TL/person from Airport to Taksim. If you want to go to Sultanahmet from Taksim, you can either take the taxi or use the public transportation. If you want to use the public transportation, take the funicular from Taksim square to Kabatas and at Kabatas transfer to tram. Get off the tram at Sultanahmet stop (or other stops closer to your hotel). This funicular/tram combo costs 3 TL/person. If you are going to stay in Istanbul at least 4 days, it is better to buy Akbil pass which is sold at Taksim square (besides Zeytinburnu and Sultanahmet tram stop
If you dont have a pick up service, you have three options, Airport Taxis, which are reliable. Airport Shuttle buses going to several main destinations ( www.havas.com.tr) and Metro, which is cheapest and often fastest. If you plan to go by Metro, there is a website, www.turkeytravelplanner.com, that provides very detailed information on how to do so--how to buy a token, how to get off at the right stop, etc.