My top tip is to avoid eateries on the ramblas - overpriced and not very good ime. Go up the Passeig de Gracia (north of Placa de Catalunya) and check out the tapas bars up there. Bar Bar Ree Bar is very good. Taller de Tapas is also good - there's one in the Carrer d'Argentario (iirc).
The Palau de la Musica is a must see. For about 8 euro you can buy tickets for a guided tour (in english every hour). It's a magnificent building and if you can get tickets to a concert there, do so!
The Seu (the cathedral) is also a wonderful sight.
The Parc de la Ciutadella is lovely for a walk on a Sunday afternoon. The zoo is there and natural history museum and you can hire one of those bike cart things to whizz about on. Or you can row on the boating lake.
Have a granissat or a horxata - there's a good place just off the Placa de Catalunya on the road that runs parallel with the ramblas.
Camp Nou is well worth a visit - it's an awesome sight even if you're not into football!
Sagrada Familia is a must as everyone else has said.
If you get the metro to the olympic village, look out for a restaurant called Marina Moncho - v v good (about 70 euro for 2 with wine).
Do use the metro, it's cheap and easy to use and air conditioned. Get a 10-ride ticket - you can use it for any number of people.