Cyclades and especially Santorini are suitable for excursions to nearby islands or even not. The islands that are closest to Santorini are Anafi, Ios, Folegandros and Milos, but you can combine your holidays with several more islands, as due to the large tourist traffic the itineraries to and from the island are in plethora.
Definitely an excursion not to be missed is in the active volcano of Santorini. Most offices organize a trip to the volcano, with a stop for a dip in the hot springs and a walk in Thirasia, an excursion that is really worth doing. There are also several offices that make private trips by boat or catamaran, offering a truly unique experience; yet the cost is quite high.