First of all, a cruise to the volcano is a "must do" in Santorini.  

There are private cruises that leave from Vlychada Port, they sail near the Red and White Beach, turn inside the Caldera, stop by the Volcano; passengers can swim at the Hot Springs and have a delicious barbecue meal on board with sea food, meat, vegetables, salads and pasta.

There are many wineries to visit, have a tour inside and wine tasting as well shared or on private basis.

The capital of the island Fira offers a variety of shops with local and designers items, different restaurants and the Archaeological Museum of Akrotiri to visit, as well as the Orthodox Metropolis.

The black sand beach is a place one needs to experience while on the island. There are as well the red (red soil) and the white(pebble stoned) beach as well.

The Ancient City of Thira is open from 08:00-15:00. The ruins of the City transfer you thousands of years back in time, before the big eruption of the Volcano that destroyed half of the island.

Horseback riding on the beach of Vlychada during sunset is a unique romantic experience.

The village of Pyrgos, on the highest point of the island with the monastery of Prophet Elias offers spectacular views all around the island.

In the center of the traditional village of Megalochori, one of the oldest villages on the island, visitors can explore the traditional architecture of Santorini and the various authentic vineyards that embrace this traditional village.
A private car tour can also be booked to take guests around the island to visit the most important sites.

For more active persons, hiking is a unique experience in Santorini.  
There is a path that goes along the cliffs from Oia all the way to Fira.

Lastly Oia village, offers one of the most amazing romantic view to the sunset.
Oia also offers many gourmet restaurants and traditional fish taverns, shops, art galleries, jewelry shops.