Sagrada Familiar, Parc Guell, Zoo & Parc De La Ciutadella, Columbus Monument, Tibidabo & Funicular ride upto the top, Montjuic and the olympic area, cable car ride down to the port, stroll down la ramblas, take in the gothic quarter and cathedral, shop down the diagonal or the grand via.
Best way to see it all is red & blue routes on the bus touristica, ignore the green route as its crap.
Just generally stroll round and take in the sights and sounds of the place, be aware of beggers and bag snatchers - there are a hell of a lot in Barcelona especially on la ramblas and its side streets, you can usually spot them as they are oldish women with babies or children who come up and ask for money whilst some little **** steals from you.
Also stop of on la ramblas and watch the 3 cups / pea game, watch all of the tourists get sucked into playing and losing 20 euros.
PS - for food go to the supermarket on la ramblas, don't sit and eat on la ramblas or any irish bars.