I figure that it was transported by cart to the sea and then by ship away from Peru. However, I understand that sailing around Cape Horn is difficult (and not somewhere you'd want to put an extremely valuable ship) and I was under the impression that pirates during the Golden Age of Piracy were primarily raiding Spanish ships, and that Spanish treasure galleons were one of their targets. Was there an overland route through South America or across Panama? Or am I just mixing up the time period?
I'm also just generally interested in the mechanics of getting silver from the New World to the Old.
It is not like it was sent straight from the mines to Seville. It was a very complex process, but I'll try to make the story short.
Silver would be mined in Perú, most notably in the mine of Potosí. The silver would get melted into ingots and into the coins. The Crown's part would need to be transported to Spain, but this was not a continuous thing. The Fleet had to be gathered and arranged in full before parting.
The Crown's part would get moved to Lima, to the port of El Callao. From there, it would be taken by sea to the Pacific Coast of Panama. There, it would by taken by land to the Caribbean coast of Panama, where it would be stored until joined by the money from the northern parts of the viceroyalty. The ships from Panama and Cartagena would then go to Veracruz, Mexico.
Veracruz was the official point of departure of the Fleet of the Treasure. The fleet would go once a year from there to Cádiz and Seville. From Veracruz, the Fleet would travel to Havana as a last stop, and then across the Atlantic.
Normally, the Treasure Fleet would be rather heavily escorted, at least by 5 heavy galleons.
This fleet system was extremely succesful, only losing to pirates and the elements some 3.5% of the ships all over 300 years. There are only two instances of a whole fleet being lost: in 1626 captured by Piet Heyn, and in 1704 captured by the British at the Battle of Rande.
Piracy was a nuisance, not quite a threat.
Sources:
Manuel Lucena Salmoral, La flota de Indias, Cuadernos de Historia 16, Barcelona, 1985.
Esteban Mira Caballos, El sistema naval del Imperio Español, Madrid, Punto de Vista, 2015.