There are so many lovely things to do! Agree with AnythingNotEverything's suggestion to avoid Las Ramblas where possible - too busy and touristy, and the little streets coming off the main drag are so much more interesting. YY to TurnOffTheTV's suggestion of Tibidabo, if you have time. The Tramvia Blau and Funicular which take you up there are great fun, and the views are amazing.
The museum of contemporary art (MACBA) in the Raval is great.
There's a fantastic cafe, Granja Viader, at 4-6 Carrer Xucla (again in the Raval), which serves the best, thickest hot chocolate ever, and has been there for ages.
In the Barri Gotic, the Museu Frederic Mares is great. It's a museum of the weird and wonderful collections of a Catalan sculptor, with a beautiful courtyard.
As Umlauf says, the Miro museum, Poble Espanyol and Montjuic castle are all very close to each other (Placa d'Espanya), and the CaixaForum (big, beautiful gallery in an amazing building with free entry) is near there too. Also in that area is Las Arenas, a former bullring converted into a mall. The shops are quite good, but the view from the rooftop is just brilliant (and there's a big Mercadona supermarket in the basement if you need juice, water, other cheap supplies).
And if you have young DCs, I bet they'd love WePudding on Carrer Pau Claris (near Passeig de Gracia). It's a cafe decorated like fairyland - all toadstools and ladybirds, and the cakes are amazing.