The only realistic daytrip possibilities from Thera/Santorini are Thirasia and Nea Kameni/Palio Kameni(volcanoes), done by excursion boats.
You may be able to engineer a daytrip to islands like Thirasia, Anafi, Ios, Sikinos, Folegandros, Naxos, Paros, by using the ferry schedules in hi-season when there are the most ferries, but it is much easier to stay at least overnight.

Santorini is actually a good starting point for backpacking around about 20 of the Cyclades Islands, you will notice a small percentage of charter flight passengers with backpacks, who will be doing just that.

Cheers for the reply Ouzo - we might look at doing an overnighter so. Which island would be closest to Santorini?