From the 4th century BC to the 3rd century AD, the area adjacent to the beach at Kionia (3km northwest of central Hora) was a major religious sanctuary devoted to sea god Poseidon and his wife Amphitrite. Now all that remains are the excavated foundations of temples, altars, fountains and baths. You can see a lot from the road but to make sense of it you really need the pamphlet that comes with admission.
Also helpful is the scale model at the Tinos Archaeological Museum.