Back when the earth was assumed to be flat, Cape Finisterre (derived from the Latin for "land's end") was thought to be the westernmost point of the world. Of course, the world moves on across the Atlantic, but Cape Finisterre is the westernmost point in Spain, and it's also the final destination for Catholic pilgrims traveling the Way of Saint James. It's about 90 km (55 miles) west of Santiago de Compostela.
If you're not making a pilgrimage, Cape Finisterre has a number of excellent beaches, including O Rostro, Arnela, Mar de Fora, Langosteira, Riveira, and Corbeiro, many of which are bracketed by breathtaking cliffs. The water isn't too cold here, making it a great destination for swimming and sunbathing.
Pilgrims can walk(!), but you might want to take a bus - a bus runs from Santiago de Compostela twice a day; the drive takes about 90 minutes. Finisterre has several quaint fishing villages to visit, as well as a number of landmarks of both maritime and religious nature. This being the terminus of a major Catholic pilgrimage, traditions abound, and if you visit the lighthouse, you can burn your boots or clothes in the steel bowls set up for this, echoing a ritual dating back to the middle ages.
While the beaches are fun and their rocky backdrops make for some striking photo opportunities, they are also the sites of numerous shipwrecks - the reef curving around the Galician shore is known for its fatal treachery, inspiring its colloquial name, La Costa Morte, or the Coast of Death.