MMMMM. I've been to Barcelona three times (one for a week, one for a job interview, one as a job trip), so I have a tiny bit of an idea. One, La Rambla, going from Plaça Catalunya to the sea, very touristy zone (if you go next weekend, you might find lots of Barcelona fans celebrating the League title at the Canaletas fountain -which actually is a normal street drinking fountain-). Going down the avenue, you'll find the Boquería market, which is a delightful place to visit (and perhaps buy something, if anything a drink in one of the pubs inside). Going from there through Carrer Ferrán, you can find Plaça Sant Jaume nearby, with the Cathedral, the City Council and the Generalitat building. And if you keep walking to the East, there is the old city, where you can find the Picasso Museum and the Santa María del Mar Church, which is a delight to see.
There is also the Palau de la Música and the Spanish Village, which is a 20's pocket-version of Spain, including streets and monuments lifted from elsewhere in the country, so tourists could see the entire thing just with a single visit to Barcelona. Also the Columbus statue at the end of the Rambla (it has an elevator inside, so you can go to the top) and the former bullring, which is now a shopping centre- I think that one was in Plaça de Espanya. If you feel like walking through green areas, there is also Montjuïc, featuring a former castle, and the Parc de la Ciutadella, with a zoo and the Catalan Parliament (don't feed the MPs!).
And, of course, everything Gaudí: Parc Güell, Sagrada Familia, Casa Battló, La Pedrera... you can't miss those.