Barcelona would be well worth a visit for the Barri Gòtic even if Gaudi and co had never existed! It has a wealth of older buildings, more indeed than in a number of places where old buildings form the only attraction.
The old cathedral has to be mentioned first. Its vast size does not detract from its appeal as a splendid Gothic building, and the streets to one side of it are narrow, picturesque and interesting. On the other side is a huge square where local people were dancing in great circles on a feast day. The interior is well worth a look and the cloisters are great.
But stay awake - I was set up for a scam here. Something must have gone wrong and I ended up OK, but not everyone has been taken to the fountain in the cloisters for a wash!
Next I will mention another church which I think is just as good, the interior at least, for the quality of its stained glass windows. This is Santa Maria del Mar. There are other churches mentioned in the Rough Guide as well, but it is not possible to do everything.
The old Palau Reial now houses the Museu d'Historia de la Ciutat. The building itself is sufficient justification for going in, but inside you can see in situ the remains of Roman and Visigothic Barcelona.
Other old lay buildings are the Ayuntament, the Town Hall (where all sorts of figures in stilts were preparing to go in procession last time I was there), the Palau de la Generalitat opposite (the home of the Catalan Provincial Government), the Museu Marès (which I defied the Rough guide by not visiting) and the Picasso museum (which is biased strongly in the direction of his early life and works, but very interesting, and again the building justifies entry).
Then on the other side of La Rambla, but still in the Barri Gòtic, are several important sights. I did not go in the Liceu although I am sure it would be good, but I bought my lunch to carry from the market every day and everybody should see it. It is a real gem, probably exaggerated by the fact that such a central market is becoming rarer. Lastly, of the older buildings, is the Antic Hospital de la Santa Creu, now used by the University. You can wander around the old garden and cloisters, sadly I saw overmuch litter when I did.
The Palau de la Música Catalana is probably in the Barri Gòtic but I include it under Modernisme.