there are beaches in barcelona down by the port olympique but they can get very busy.
just outside the city you have a number of beach areas costa del garraf, costa dairada, costa del marasme, costa brava further up the coast.
your in the gothic area so you will love la ramblas (the market is on a sunday), if you like to party dude it is a party town, things do not start even warming up till well after midnight.
things you must see, do not leave barcelona without going to PARK GUELL gaudis park and the SAGRADA FAMILIARE the unfinished cathederal.
be careful while walking around some parts of the gothic quarter ask at your hotel the places to stay away from.
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i have travelled the world and i love barcelona, you will love it too i'm positive of that! have a good one!