Forget Turkey. You don't have nearly enough time.

You basically have two options to get there.
1. Fly or ferry back to Athens, and then fly on to Istanbul. This is going to expensive, and will cost you at least one day in transit.

2. Ferry from Santorini to one of the Greek Dodecanese islands (Kos, Samos or Rhodes) and then ferry directly to the Turkish coast. Again, time is your big enemy. From Santorini these islands are at least one day away, and you will still need to take a separate ferry to the Turkish coast. Additionally, you will end up in on the Aegean coast...not Istanbul.

Find an island with good beaches, and the kids will have a great time. Santorini has very poor "volcanic" beaches. Local islands with wonderful beaches are Naxos, Paros, Folegandros, and Milos. All are within a couple of hours by ferry from Santorini..