I recommend a restaurant called Tragaluz up down a small alley off Passeig de Gracia - always reliably nice,stylish, easy, yummy food.
There are lots of small tapas bars in the streets running parallel to the seafront (turn left when you get to statue of Columbus at the bottom of the Ramblas). The food will be good but the service is often surly. If you are up in the Eixample, try Cervercia Catalana off Rambla Catalunya - maybe on C/ Provence or Mallorca - fabulous tapas and very popular with well-heeled locals (but it's not expensive). Or try Sunday lunch - seafood & cava - in Barceloneta - old fisherman's quarter on the beachfront before you get to the Hotel Arts.
The market half way down the Ramblas on the right hand side is the Boqueria - worth visiting - amazing.
Shopping - Zara & Mango cheaper than here s worth a look. For more expensive stuff try Purification Garcia - v. nice.
I'd recommend going up to the Miro museum. You can go on the funicular and the views from the roof terrace are amazing.
Gaudi stuff - can't remember the building on Passeig de Gracia but it has a good museum and again, lovely views from the roof. It's on the right hand side quite far up (away from Placa di Catalunya).
My dh's brother lives there and we used to go over 3/4 times a year till the arrival of dd1 - it's a fabulous city so have a great time. One word of warning - there is quite a lot of street crime and I'd avoid the Ramblas and small streets around that area at night.