Loads of Barcelona questions & answers, so be sure to also do a search "Barcelona".
I've always stayed on or just off Paseig de Gracia, towards Diagonal, and love it. Easy access to lots of restaurants, barrio Gothico etc. El Born is also trendy with lots going on.
If it's your first visit, you have to go Sagradia Familia, Barrio Gothico, parc guell, Las Ramblas ( watch pickpockets,), Mont Juic Olympic Park, and if you have time, consider a day trip to Montserrat or Girona. almost 10 years ago