If you have any problems with vertigo, DO NOT go up the spire of Sagrada Familia.
If you don't mind being a bit touristy, make use of the hop-on-hop-off bus tours that go from the centre of the city. They run three circuits, and the north one is a very good way of getting to see all the sites. The south one is quite good, but not as good.
And it goes without saying, go to the Nou Camp and take the stadium tour. It's worth visiting the Olympic stadium too, for the views.
Sorry that my stuff is all touristy, but I only had two days there and that was all we had time to do.
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