Hydra Town. The main village is as cute as they come, and you can easily spend a few hours browsing through the shops and having lunch near the marina. If you like to take an easy dip in the Med without taking a water taxi to the beach, walk around the promenade and up the hill to Spilia, a beach bar that also has a swimming platform. There are also nice views here.
Beaches. As with most Greek islands, Hydra has some beautiful beaches. Vlychos and Mandraki are two that have tavernas and beach chair rentals; to get there, you'll have to take a water taxi from Hydra Town.
Walking. If it's not too hot, there's a nice coastal path that runs from Hydra Town to Plakes. There's a hotel here called Four Seasons (no relation to the luxury chain) where you can rent a beach chair and enjoy lunch.