Transportation in Istanbul is very easy, if you are in the city center. There are several options like metro, bus, tram, small bus, metrobus etc. It is better to use metros and trams rather than buses if you are close to those, in order to skip the traffic.
We use a card called Istanbul Card, which you can buy for 10 TL (3 EU) and charge it to use. It is good for public transportation. You will be paid back, when you return the card.
By following this link, you may find a very good explanation for the transportation system.