There is a little known Gaudi building called Palau Guell just off the Ramblas.
Most Gaudi's house are more beautiful from the outside than inside. So yes you can visit any of them, but if you are short on time, I would just walk around and get a feeling of the city, admiring the buildings from the streets.
The old city, with its narrow streets, or take the cabin across the harbour to the Barceloneta , walk from Plaça Espanya to Mont Juic. There are so many options on what to do.
The Sagrada Familia is certainly unique but it is quite out of the way and unless you go there early morning, it is really crowded and you can't really take the time to observe the infinite details that makes it unique.
When you are are the restaurant, keep you bag on your lap and just put your napkin on top of it, never hanging from the chair or on the floor.
What makes barcelona Barcelona is not so much all the beautiful buildings but the people who live there. Happy, relaxed, sitting in bars at any time of the day, kids playing soccer in Gracia's squares at midnight on schooldays, the hidden gardens behind buildings in which you can sit on a bench and eat a "bocadillo" .
I live now in Sydney and still after 4 years miss Barcelona dearly.