Superunknown and Badmotorfinger are really all you need to start off. Their other albums, like Down On The Upside are very different albums, and are an acquired taste. Superunknown would be my favourite Soundgarden (the band name is one word by the way), with Black Hole Sun, Spoonman, Superunknown, My Wave, Fell On Black Days. Badmotorfinger, like Ev0 said, is definitely Chris Cornell's strongest vocal performance, and contains one of my favourite Soundgarden tracks, Jesus Christ Pose. Rusty Cage and Searching With My Good Eye Closed are also some of the highlights from the album.
But definitely look into those before looking anywhere else.
