Lots of great bands (and offshoots/side projects) listed in this thread. Some others to try:
Green River (Jeff Ament & Stone Gossard - now Pearl Jam, Mark Arm - now Mudhoney)
Malfunkshun (Regan Hagar, Andrew Wood (deceased) - former roommate of Chris Cornell)
Mother Love Bone (Ament & Gossard, Wood)
Temple of the Dog (one-off tribute album to Andrew Wood, essential what would become Pearl Jam, plus Chris Cornell)
Satchel (Shawn Smith, Regan Hagar)
Pigeonhed (Shawn Smith, Steve Fisk)
Three Fish (Ament side project)
Brad (Regan Hagar, Shawn Smith, Gossard)
Also worth checking out is Mind Funk - their second album, Dropped, is amazing. Really dark.
It's also worth catching this (
http://www.imdb.com/title/tt0105415/) for a performance by AIC, appearances by Pearl Jam members, a very funny cameo from Chris Cornell, and lots of subtle references to obscure Seattle bands (keep an eye on the graffiti).