Hi -we have had 3 separate days in Barcelon so far, and still haven't seen it all! (I think we're a bit slow!)
Sagrada Familia (new cathedral) is the top site- it gets very busy, but there is more to see outside than inside.
Parc Guell- fantastic place, a bit further away, but you just HAVE to go there.
Pueblo Espanol- a full size village of all different styles of Spanish architecture, craft shops, cafes, restaurants, art gallery etc. very quaint.
Old cathedral and old town, lots of tapas bars, lovely shops, markets.
The Ramblas - long pedestrianised walk with flower stalls, bird stalls, street entertainers.
And so much more, e.g. fountains, Gaudi buildings, etc.
The tourist bus goes to all the sites, and is reasonably priced, but if you want to go direct to a site you may spend too much time on there going on its roundabout route.