Barcelona is so big that I would advise a person who is only staying for a weekend or a week to take the bus tours. There is a blue bus and a red bus. The service is excellent and you can get off the bus whenever you like and visit that attraction, and when you are finished you just hop on the next bus that comes around. The Gaudi park, cathedrals, Olympic Stadium, Christopher Columbus statue, Monserrate (an old monastery), and the new Camp Football stadium are just a few of the most popular sites in Barcelona.
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Taxis are cheap, and for getting around the city the metro is brilliant.