There are trails all over rocky Anafi. Terrain (http://terrainmaps.gr) produces an excellent hiking map for the island.
The 7km walk to Zoodochos Pigi Monastery along the south coast (around 2½ to three hours one-way) is a rewarding expedition, but it’s a fairly tough trip in places and a day’s outing there and back (consider taking a bus to the monastery in summer and walking back).
Once you're there, the island’s most celebrated trail is the rocky ascent from Zoodochos Pigi up Kalamos to Moni Kalamiotissas at the top. The walk is 2.5km with 400m of elevation gain; allow an hour or so each way.