one catch with photographing famous places it that they all have their famous shots - sunset on delicate arch, yosemite valley, the golden gate bridge from the marin headlands - they're the pictures that you see over and over and over. if you visit these places at the right times you can usually see a row of people with their tripods set up to capture that famous shot for themselves.
i'm not against the idea of taking famous photographs for yourself, in fact i think it's a good place to start - just don't end there.
here are two familiar shots of brooklyn bridge that i took in december:

the straight on shot of one of the the towers...

the shot from brooklyn with downtown manhattan in the background...
now here's a shot that i took as i was crossing the brooklyn bridge. i guess it's not really a picture of the brooklyn bridge per se but rather a deeper view into manhattan which includes the brooklyn bridge (as well as the manhattan bridge, the empire state building, and the chrysler building).

the world is full of new ways of seeing things if you just take a little more time to explore and find them.