What you mean and ?.... with a week you won't be able to see but the surface. The city has a lot to offer and you won't finish it in a week -unless you're only interested in ticking off a list the handful of sites that are repeated over and over in many tourists guides (as if there was nothing else :)... that you can do in 4-5 days.
Yet if you want to alternate and do some escapades, there are wonderful rural and coastal towns that are just a (short) ride away... Barcelona is the capital of Catalonia, a region the size of Maryland that has miles of beaches, several national parks and marine reserves, some of the highest peaks in the south of Europe (in the Pyrenees).... and several thousands of years of history so you've got remains from all eras: Iberians, Phoenicians, Greeks, Romans, Visigoths.... and since the birth of this small nation in the 9th century to today, all sorts of architectural wonders, folk festivities and traditions that are regularly shown in festivals all over Catalonia. Have a look at www.catalunya.com and www.infocatalonia.eu to have some glimpses. Something you ought to take into account is that there are plenty more things to see besides cities.
