Strolling through the streets of Barri Gotic (Barcelona's old town) gives you a true appreciation of history, with a melange of medieval buildings and lovely cathedrals mixed with great modern fare like ubiquitous tapas restaurants and tiny cafes. You could spend a day exploring every niche of this neighborhood and its many plazas (of which Placa Reail is a lively highlight) and basking in well-preserved antiquity.
The majority of businesses are southeast of Santa Eulalia Cathedral, an area worth exploring for an afternoon or evening just to wander the medley of pedestrian avenues. You'll find a variety of Spanish restaurants (especially those that cater to the tapas crowd), as well as a surprising number of Japanese/sushi places that are valued highly by the locals (Temaki-ya and Sushi Ya 2). There are plenty of opportunities to grab a drink as well: get your indie vibes at Bar Manchester, your craft beers at La Cerveteca, and your gorgeous interior design fix at Milk. Also, hit up the aforementioned beacon of socialization Placa Reail (located very near Las Ramblas) for a selection of high-end nightclubs like Jamboree and Karma.