We last went in November, stayed just outside the Gothic Quarter. We always use the bus service. Think it was about €6 each from the airport to Plaça Catalunya (Square of Catalonia). We use the buses to get about and even use one of the tour bus companies, buy a ticket and you get use their buses all day long, stop outside the main gate at that "tiny" stadium!
Barcelona is fairly spread out but most places are within walking distance, just get a good map/guidebook. AA guide book is brilliant for everything, from places to eat, sight to see and how to get there.
Enjoy it, not my favourite city but a truly vibrant place if you go down the Las Ramblas at night time. Just hold onto your wallet and don't get sidetracked by the street entertainers (as they will take your money, literally).