What would be the best way to get to the Greek islands to Istanbul? Is this easily feasible or will it be pretty expensive. I have just heard that Istanbul is an amazing city and I really want to go!
There are several ferry routes into Turkey from the islands, it really just depend on your island itinerary. There are 5 Greek islands from which you can enter Turkey: Lesvos, Chios, Samos, Kos and Rhodes. You'll have to do some research on your own to decide which Greek islands you want to visit, and after you've figured that out by looking at a map of the Aegean Sea you'll know which of the 5 islands you will be closest to for the ferry to Turkey. Once you know that you can start to figure out how to get to that island for the crossing to Turkey, and then how to get to Istanbul from there. Buses are cheap and frequent in Turkey. They're also modern, comfortable and air-conditioned, which is a blessing in the hot summer months.
I had planned a trip the opposite way (unfortunately it has fallen through) that, for you would be a ferry or flight to Samos, Greece; Ferry to Kusadasi, Turkey; see Ephesus; then a budget flight from Izmir to Istanbul. Buses are a good way to travel in Turkey, but watch the distance, or you will find yourself travelling all day or overnight. Budget flights in Turkey are becoming more common and reasonable.
Keep in mind that the ferries between the Greek islands and Turkey are seasonal--most stop running in early fall and don't pick up again till late spring.
Here is a good website to help you make that determination: