Start at the Cathedral of Santa Eulàlia on Plaça de la Seu. The body of the building is true Gothic, thirteenth to fifteenth century, raised over the Roman city of Barcino; but the main façade you see is a late nineteenth-century neo-Gothic addition, begun in 1887 and completed, with its central spire, only in 1913. In the cloister, thirteen white geese live among the orange trees and the fountain, one for each year of Santa Eulàlia's life — she was Barcelona's co-patron, killed at thirteen in 304 AD, and the geese are the city's softest memorial.
From there, wander rather than march. Plaça del Rei opens onto the royal palace complex, with the Torre del Rei Martí and the Royal Chapel of Santa Àgata, and beneath the square the Museu d'Història de Barcelona takes you down into excavated Roman streets, workshops and a wine-making plant. A few steps away, the Pont del Bisbe arches over Carrer del Bisbe — one of the most photographed views in the city, and a total fake, built in 1928 as neo-Gothic pastiche for the 1929 International Exposition.
Then let yourself get lost in El Call, the medieval Jewish quarter, where the streets narrow to a handshake's width and the city's Jewish community lived and worked until the expulsions of the late fifteenth century. Plaça de Sant Felip Neri closes the loop: quiet, cool, with a fountain and church walls still pockmarked by shrapnel from a Civil War bombing in 1938.
Notes
Comfortable shoes with grip matter here: these are polished, uneven stone lanes that turn slick in the rain. The lanes are busy and pickpocketing is the local sport in the tourist core, so keep phones and wallets in front pockets and bags zipped across your body — the same advice applies on La Rambla. This is also the easiest place in the city to lose your group; agree a meeting point (the cathedral steps work well) and remember the quarter is small enough that you will find each other quickly.