I would recommend you to stay in Gracia or Barceloneta neighbourhoods. Both have a great daily market for fresh foods and lots of little stores for various needs.

Gracia is an attractive barrio with narrow streets and small squares, popular with young expats but not touristy and a bit bohemian. There are lots of cinemas, theatres and restaurants.

Barceloneta is a fantastic place 2mins from the beach and a lovely 10-15 min walk to Las Ramblas. Bear in mind that you are living amongst the locals in converted fishermans dwellings with people going about their daily business. There are lots of seafood restaurants and tapas bar and you can enjoy clubs and lounges in Pº Maritimo de la Barceloneta, Borne area or Gothinc quarter.

Have fun!