Barcelona rewards a week, but with only one day you have to be ruthless: pick Gaudí's icons and the historic core, and let the rest go. This itinerary deliberately skips the beaches, Montjuïc's museums, and most of the smaller churches so you can see the essentials without racing the clock.
Book a timed entry to the Sagrada Família for first thing in the morning, when the light through the stained glass is best and the lines are shortest. From there, take a taxi or the metro up to Park Güell for a mid-morning walk through Gaudí's mosaic terraces and the city views over Barcelona. By midday, head back down into the Eixample district to see Casa Batlló and Casa Milà (La Pedrera), two more Gaudí landmarks just a few blocks apart on Passeig de Gràcia — a quick look at the facades is enough if you're pressed for time. Spend the early afternoon in the Gothic Quarter (Barri Gòtic), starting with Barcelona Cathedral (La Seu) and its cloister, then losing yourself in the medieval lanes toward Basílica de Santa Maria del Mar in the neighboring El Born district. As evening approaches, wind back through the Gothic Quarter for tapas and a stroll past the Roman-era corners of the old town before calling it a night.