Take the tram (T1 Bağcilar merkez-Kabataş) to the Kabataş end of the line. People coming from Taksim, would come through (F1 Taksim-Kabataş) Füniküler. When you disembark, you have a choice of two ferries.

To the left is the faster of the two which for 5L takes 1h15m to get to the large island (Buyukada). A representative from this company will likely be near the tram exit directing tourists to this boat while suggesting it is the only way to get there.

However to the right is the more popular ferry with locals which is only 4L but takes 2h and stops at the smaller islands along the way. On the cheaper ferry, if you are not near the front of the queue, there is a chance you will have to sit on the floor or stand for the entire trip. There does not seem to be an enforced capacity limit.  

 Refreshments are available on both ferries.