The Barri Gothic is the heart of the old town. Enclosed inside the Roman walls, you can find Barcelona's oldest buildings placed in a labyrinth of small streets where you can imagine being in the Middle Ages.
We can begin the tour at the Plaça Nova, and old marketplace where you can see the Palau del Bisbe flanked by two towers of the old city wall. Opposite there is the Casa de l'Adiarca, currently an archive.
Next to it, there is the Plaça de Sant Felip Neri, a small, quiet square with an offbeat museum, the Museu del Calçat (museum of footwear!)
Now, you can enter the Cathedral through the Porta de Santa Eulàlia and access the cloister, with its palm trees, magnolias and medlars, among which you can see the thirteen geese that live in the cloister (it's said that there is a goose for every year of Santa Eulàlia when she was martyred).
Some parts of the cathedral are as recent as late XIX century (like the western façade) while other parts date from XI century (like the entrance to the chapel of Santa Lucia and the Porta de Sant Sever).
After walking through the cathedral itself, you can visit the Cathedral's museum and see the gothic tabernacle and the throne of King Marti I.
In front of the cathedral there is the Plaça de la Seu, where you can buy religious memorabilia. If it's near Christmas, you'll have to thread your way through the crowd buying figures for nativity scenes, some of which can be very disconcerting for a tourist, like the boy shepherd moving his bowels (known as "el caganer").
Following Carrer dels Comptes you arrive to the Museu Frederic Mares, which shows the artisits sculptures, a selection of sculptures from various ages that he restored, and a collection of memorabilia.
From there, you'll arrive to the Plaça del Rei with two interesting sites to visit, the Saló del Tinell, centre of the Catalan kingdom in the XIII, when its ships ruled the Mediterranean and where the Reyes Católicos received Christopher Columbus back from America; and the Museu d'Història de la Ciutat, where you can see the foundations of several Roman buildings and streets.
You can end your tour following the Carrer de la Llibreteria until the Plaça de Sant Jaume, the old Roman forum, with the Palau de la Generalitat (house of the Catalan government) at one side and the city hall at the other.