Perhaps surprisingly in such lunar terrain, there are two spectacular hiking routes on Santorini:

From Imerovigli to Oia on a recently improved and signposted trail skimming the caldera edge (allow 2 hours).

From Perissa to Kamari, along an old cobbled, hillside trail, via the ruins of Classical-Roman Thera and the cave of Zoodóhos Piyí, containing the only potable (and very tasty - fill your bottle) spring on the island (1hr 30min). In Kamári, shuttle boats await to whisk you back to Périssa.