Gothic quarter is beautiful - some really old churches and little squares. Lot of small restaurants here which may not be in a guide book...don't be afraid to try them.
PlaÃ§a del Pi that is shown in the Gothic quarter video has Bar del Pi where artists hang out. A local Barcelona arts market is also organized here.
I love the Gothic Quarter, lived there for 3 months...love it. Check out a bar called Pile 43. They have all 70's furniture and fixtures, and they are all for sale. Not to mention the best mojitos i've had anywhere.