El Barri Gòtic (the Gothic quarter) is the medieval quarter of Barcelona. I recommend that you spend time just walking around in the maze of narrow streets and explore the many squares: I love the Plaza Reial, for example as it bears a kind of splendor when you enter through its arched gateways. The neoclassical facade surrounds the many palm trees on the square. Located just off the Rambla it is usually a little quieter, although late at night there are often a few drunk tourists who end up celebrating their vacation loudly in the fountain at the center.
Around 9pm, a line starts forming halfway to the center for the restaurant Les Quinze Nits. It is pretty decent, and despite the long line, the wait is not that long as they have people enter by groups of 20 people or so. My definite favorite hang out on the Plaza is Tarantos, where you can experience some excellent flamenco in a tiny room for 6 Euros + the cost of a drink inside, and it is a different show every night. It lasts about 45 minutes, but it is laid back and so much better than the big touristy shows that they try to sell you on La Rambla for 50 Euros.
Another main square in the Barri Gotic is the Plaça Sant Jaume, where the City Hall (or Ajuntament) stands proud. It seems like every street from the Barri Gotic merges onto this square. Apparently it is where the locals demonstrate and is basically an important meeting place. I was there during Christmas 2006 and they had a huge beautiful nativity in the middle, but I believe that they have that every year. Something to see.
The Plaça del Pi is interesting too, especially on Thursday morning when they have a little market of local goods such as cheese, meat, and cakes. On Calle Bisbe you cannot miss the "bridge of sighs" which is quite pretty and photographed but which was also apparently built in the 1920's... Quite a disappointment, but still a nice picture.
Don't forget to look up to see the gargoyles and gorgeous facades.
The Cathedral of Santa Eulalia certainly is a highlight of the Barri Gotic. You can enter for free in the mornings and late afternoons to visit the church and the cloister, however the rest of the time is dedicated to worshipers and ticket holders so plan your visit. Inside you can see the tomb of Santa Eulalia, a 13 year old girl and saint patron of Barcelona who, in 303 A.D, was said to be rolled down a hill in barrel full of broken glass and nails for protesting to the governor against his cruelty towards the Christians. Behind the Cathedral is the Baixada de Santa Eulalia, the slope where the horrifying event took place.