My 2003 came with Bridgestone Dueler H/T's I believe. They perform better, overall that my friend's 2000 RX which has Goodyear tires. The RX tracks straight w/o need for minor steering corrections; the tires are quiet. During last year's snowstorms in the NE, I never got stuck, but they don't seem to have as much snow traction as the Goodyear's. The traction and stability control on the 2003 helped with that though. If I were looking to replace my tires now, I'd get the Duelers again.
As for larger/wider tires, check with Lexus. Tracti>Original tires are getting worn down and am looking for new ones. Any