Hi - don't know about the cost, but yes you can visit other islands. we did it in reverse. Staying at Paros and visited Santorini, there were also people on our boat staying on Naxos who visited Santorini also. There is, I am sure, others nearby such as Ios which you can visit
have a great time - Santorini is breathtaking although we only saw Thira and Oia
Heather

Yes its simple to do it ! You can visit most of the nearby Cycladic islands for a day - or if you like it - stay over & make a trip of it ! Check the flying dolphins website (if they have one ) as these are faster boats than the inter-island ferries . You'll find they all run most days there & back & you can buy the tickets at travel agencies or at the port - Athinios (it's miles to get down there on a zig-zag road with some hair raising twists & turns , so you will need to get a bus or taxi )
They are not expensive and you will need to be at the port about 1/2 hr before it's due to go.
We went to Naxos (about 5 hr trip ) once & then another time to Ios , takes about 4hrs.

Thanks for both your replies

I have had a look at the flying dolphins website, that has been very helpful. The prices are reasonable. If anyone else would like to have a look the web address is http://www.dolphins.gr

It's not possible for most of the nearby islands only Ios, Naxos, Paros, Thirasia. There are Hi-speed catamarans and ferries the Hi-speeds are 2x the speed and price. The first ferries leave Athinios around 08:00.