I spent four months on exchange on Barcelona this past fall, and loved it. Agreed that 4 days is enough if you're trying to hit a few other places on your trip. Feel free to PM me if you'd like any more info, but I would recommend staying in the Born or Gothic Quarter.
As for sights to see, one of my favourites was Mount Tibidabo and Park Guell (you need tickets 1 or 2 days in advance). I'd skip Montserrat, it's nice but also a mountain and we have plenty in Canada.
The picket pocket thing was a little overblown, as the previous poster mentioned keep your wallet in your front pocket and youll be fine. I was robbed while I was there though (not pick pocketing), so I'd recommend not walking alone between 3am and 5am on quieter streets or near the beach.
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