Depending no how far north you go, there are plenty of great areas outside of the central core where you can walk, explore and delve into nooks of the city that aren't as greatly known.
Just outside the center is the Ribera neighborhood. This is a section between the Born and the Eixample and to many is considered to be slightly away from the action, yet still close enough to feel like you're a part of it. The Ribera is known for it's classy tapas joints, wider streets and modernist architecture and is a great place to rent an apartment if you're staying for a long time and would rather not stay in a hotel.
Another part of town worth visiting is Glories. Close to Poble Nou, this neighborhood is an extension of the up and coming 'Silicon Valley' and ripe with young energy and a cool new vibe. There is a great shopping mall in Glories, which is less busy and has more space than the ones closer to the city center.