I stayed at the Casa Gracia Hostel (casagraciabcn.com) which is quite central and close to public transport (Diagonal tube station) and it was alright, but apparently it has really changed, according to the pics on the website... (to the better)
You should definitely go up Montjuic, you get a stunning view over Barcelona! And the touristy stuff like Park Güell, Sagrada Familia, Las Ramblas, El Mercat de la Boqueria (you can get good fruit and smoothies for cheap), Plaza Catalunya, Barrio Gotico, Barceloneta Beach... is really worth it!
And always look out for pickpockets, there really are a lot of them in Barcelona...