Metro and railway system is fairly developed in Istanbul, but not enough to reach every corner. They can be used for main lines, but you would need cabs and buses elsewhere. Also, people tend to drive their own cars; causing traffic jams. So I’d recommend you stay out of traffic in rush hours. Taksim Square is the center of public transport, you can find any transport anywhere.
Cabs are relatively cheap in Turkey. They should be preferred for short distances. Also, please note that there is NOT a surcharge in night time. I would like to warn you about the drivers; noticing the passenger is a foreigner, some of them tend to take the long way. Fares above TL100 or USD50 would be abnormal.
Transport from Ataturk Airport: Ataturk Airport is the main airport of Istanbul. It is on the European side, and closer to city center. There is a subway line from the airport, but you have to change several lines to get to city center. Considering your luggage, it would be wiser to take the shuttle for approximately $8 per person, or to take a cab (TL50-60 or USD25-30 in day time to Taksim Square depending on traffic). If you will be staying on the Asian side, you have two options: either shuttle or cab. The shuttle would be much more cheaper (TL10 or USD5). Click here for the shuttle routes and rates.
Transport from Sabiha Gokcen Airport: Sabiha Gokcen Airport is the secondary airport of Istanbul, and is on the Asian side. Just take the shuttle for TL12 or USD6. Click here for the shuttle routes and rates (Little information on the name of the airport: Sabiha Gokcen (Sabiha Gökçen) is the first Turkish female aviator and the first female combat pilot in the world. She was one of the eight adoptive children of Mustafa Kemal Ataturk).
The main subway line is the one between Taksim Square and Levent. It is on the main commercial and residential area of Istanbul. If you are planning to go somewhere around the subway line, definetly take the subway to avoid traffic.
Ferries are also an important (and fun) way of transport in Istanbul, so I’d strongly recommend you to use them. To travel between European and Asian sides, just take the ferry. It is absolutely more fun, and faster. Do not miss this chance, enjoy the breeze of Bosphorus and the magnificent view. You may also consider taking a scenic Bosphorus ferry tour to enjoy Istanbul more, where the ferries travel all the way through Bosphorus, both by the European and Asian banks. Click here for details. You may want to experience to drive from one continent to an other, and the view from the bridge; you may take a bus as well. Take the bus using the Bogazici Bridge (BoÄŸaziçi Köprüsü) a.k.a. “The First Bridge” (“Birinci Köprü”). Click here for ferry routes and timetable.
If you are planning to use public transport (Subway, buses, ferries, etc.), buy an IstanbulKart – a prepaid RFID card valid for all public transport (There is an other type of prepaid device called “Akbil”, you can use it as well the same way as the IstanbulKart). They provide nearly 10% discount for each ride, and also nearly 50% discount for transfers (Transfers are valid among all public transports, e.g. from subway to ferry). You can buy IstanbulKart or Akbil from the main station in Taksim Square. If you do not have an IstanbulKart or Akbil, you need to buy tokens for metro and ferry; or take private buses (“Özel Halk Otobüsü”) that you can pay in cash as well.
Fares for public transport (valid as of March 2012): Fares may vary depending on the length of the route, or if the line travels across continents; but mainly in city center it is TL1.75 or USD1 (with IstanbulKart, else TL2 with tokens). If you make a transfer using IstanbulKart, it is TL1. You need to pay TL6 for IstanbulKart, and to add credits. When needed, you can add credits using the machines in subway stations.