If I were you, I would go through the Old City slowly, seeing the Cathedral, las ramblas, Plaza Jaume I, Ciutadella, etc.
Then you would want to see the Modernist things, ie Casa Battlo, Sagrada Familia, Palacio de la Música (incredible).
The Museo Nacional and the Museo Picasso are worth a visit. Also any guide book will be full of other sights. I highly recommend the Pedralbes Monastery which has an amazing gothic art collection. It's accesible by public transport (
www.tmb.net). If you have time Montserrat as mentioned previously is well worth the trip.
Half of the appeal of Barcelona is that it's a very pleasant city to walk around, and one can spend hours just doing that.