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Monte Castillo is riddled with caves containing Paleolithic art about the same age as Altamira. Two caves are open to view, and here you see the originals not a replica. But that means tightly restricted numbers, so you need to book ahead, you're unlikely to find availability on the day. It's worth seeing both: Castillo, and Las Monedas named for coins found there.
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Museo Nacional y Centro de Investigación de Altamira
The caves of Altamira, discovered in 1868, have prehistoric paintings as old as 36,000 years - they were added to or amended over another 20,000 years. The condensation from 20th century tourists damaged the paintings so the caves were closed to the public and a replica cave built - this is what you tour, along with the adjacent museum. The true cave is 500 m north but closed off. The tickets can be bought via the official page or on the spot - but in the latter case, expect you may be waiting 3h or more get in.
Cistercian Monastery / abbey built in neogothic style 1906/10 and now run by their branch order the Trappists. You can attend mass and buy their famous cheeses.