Kardamyli (15km north) is the most rewarding short trip. The old village has a beautiful stone Ktenas tower-house complex, a Byzantine church of Agios Spyridon, and good tavernas. The gorge of Vyros, above the village, is one of the best hikes in the Peloponnese — two gorges converging, with plane trees, springs, and abandoned chapels. Patrick Leigh Fermor lived in Kardamyli for decades and wrote about the Mani here.
Diros Caves (35km south near Pyrgos Dirou) are one of Greece's most spectacular natural sites — an underground river system explored by flat-bottomed boat. The stalactite formations are extraordinary. Open daily; booking ahead is advisable in peak season.
Mystras (60km northeast) is a ruined Byzantine city on a hillside above Sparta, a UNESCO World Heritage Site with frescoed churches, palaces, and fortifications. A half-day trip. Kalamata (45km north) has good food and a pleasant old quarter; the Benakeion Archaeological Museum is worth a visit.