What is it? The National Art Museum of Catalonia, where you can get an excellent overview of Catalan art spanning from the 12th to the 20th centuries.
Why go? For the impressive Romanesque collection with 'rescued' 10th-century murals, the fantastic gothic collection, and the uniquely Catalan, Modernista collection. Plus, once you've climbed the steps up to the former castle, you've earned that stunning view of the city below.
Top tip: Looking for something low-key in the evening? Head to Gresca. This cosy bar is a short distance from the museum and has an affordable menu and a great selection of natural wines.