For history & culture, Hospital de Sant Pau (Recinte Modernista) delivers.
Domènech i Montaner's mosaic pavilions, vaulted tunnels and garden walkways make this the most underrated Modernista site in the city, and it's a UNESCO site that almost nobody queues for. You can wander the grounds slowly and actually photograph an empty arch. The catch: it's a 10-minute walk from Sagrada Família and people rarely budget time for it, so it gets skipped.