I've been through Barcelona twice (during the same trip) and was in a similar situation to you, not really knowing what to expect.
To be fair, you'll find a lot just wandering around. The around the centre, big market and along the harbour there's plenty to see.
Another good shout for us was the bus tour, which takes you around various routes and stops at numerous places (like La Sagrada Família, nou camp) where you can jump off, explore then jump back on the buses again after. It was very reasonably priced too