I am a “local” in barcelona, have living here on and off for arround 3 years.
First of all: Airbnb is really expensive in this city. I have a good friend with a lot of apartments for large groups and this is a real business here.
Locals normally spot apartments on a website called “loquo”, but it can be difficult if you don’t speak spanish and if its for very short rentals. There are specialized agencys for short term rentals like http://www.apartmentbarcelona.com/ and http://www.ainb.net/. I’ve made really good experiences with http://lodgingapartments.com/, they take a comission but have really great, furnished apartments in barceloneta.
For the best quarters: I lived in the “Born”, its not quite as turistic and “dirty” than the gothic or raval quarter, but still right in the city center. A lot of expats, students and freelancers life there and it has a great atmosphere with a lot of bars, small shops and art galleries (although its changing). If you like to be a little bit more outside the city center gracia is awesome as well, a lot of the alternative party scene is happening there.
If you have any questions just let me know