You could easily spend 5 nights in Barcelona, and still not see/do everything that you want to.
As Tabulah says there are some lovely beaches just outside Barca,
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The tourist bus is a great way of navigating the city, and has 5 or 6 routes that cover all of the main tourist areas/attractions, as well as going out to the beaches.
El born is a fab area, very arty and boho?
There is a lovely bar next to the wax works museum done out like a fairy grotto.
There is an artists village somewhere on the tourist bus route, i forget what its called, but its lovely to walk around for a few hours
Cable car to the to the top of the mountain/out over the harbour
Nou Camp is worth a visit even if you arent massive football fans
The marina development is gorgeous, and you could easily spend a day down here, boat trip, lunch, people watching, shopping, ice cream, watching the cruise ships come in.
The Font du magic is a water and light show that is on most nights, its amazing, totally worth a visit, although depending on where you are staying probably easier to get a taxi than walk... much nicer in the dark though.
The zoo is lovely, as is the aquarium, although if I was on a child free break then I would avoid like the plague
Placa Reial is a beautiful square just off la Rambla is nice for a drink/people watch
Second Park Guell, although we went to the Sangrada Familia and were very unimpressed.... huge touristy building site, and I am sure Gaudi would be turning in his grave if he could see his beautiful church complete with vending machines and super speed lifts.
Have a fab time, as you can probably tell its one of my favourite places