Here's the order I read them in:

  I, Robot (1950)
  Foundation (1951)
  Foundation and Empire (1952)
  Second Foundation (1953)
  The Caves Of Steel (1954)
  The Naked Sun (1957)
  The Robots Of Dawn (1983)
  Foundation’s Edge (1982)
  Foundation and Earth (1986)
  Robots And Empire (1985)
  The End of Eternity (1955)

I haven't actually finished the final two because it took me a while to find Robots And Empire. I'll have to reread from the beginning.

Here’s Asimov’s recommended order. This one is pretty much chronological for characters, I think, which would spoil some narrative twists. Kind of like how watching the Star Wars prequels first would spoil The Empire Strikes Back.

  Either The Complete Robot (1982) *or* I, Robot (1950)
  The Caves Of Steel
  The Naked Sun
  The Robots Of Dawn
  Robots And Empire
  The Currents Of Space (1952)
  The Stars, Like Dust (1951)
  Pebble In The Sky (1950)
  Prelude To Foundation (1988)
  Forward the Foundation (1993)
  Foundation
  Foundation and Empire
  Second Foundation
  Foundation’s Edge
  Foundation and Earth

And here is an Expanded Chronological order, which includes books written by other authors after Asimov died:

  I, Robot
  The Rest of the Robots (1964)
  The Complete Robot (1982)
  Robot Dreams (1986)
  Robot Visions (1990)
  The Positronic Man (1992) (co-written by Robert Silverberg)
  Mother Earth (1948) (minor inconsistencies)
  The Caves Of Steel
  The Naked Sun
  Mirror Image (1972)
  The Robots of Dawn
  Mirage (Mark W. Tiedemann) (2000)
  Chimera (Mark W. Tiedemann) (2001)
  Aurora (Mark W. Tiedemann) (2002)
  Have Robot, Will Travel (Alexander C. Irvine) (2005)
  Robots And Empire
  Caliban (Roger MacBride Allen) (1993)
  Inferno (Roger MacBride Allen) (1994)
  Utopia (Roger MacBride Allen) (1996)
  The Stars, Like Dust
  The Currents Of Space 
  Pebble In The Sky
  Prelude To Foundation
  Foundation’s Fear (Gregory Benford) (1997)
  Forward the Foundation (MINUS the final section of the epilogue)
  Foundation and Chaos (Greg Bear) (1998)
  Foundation’s Triumph (David Brin) (1999)
  Foundation
  Foundation and Empire
  Second Foundation
  Foundation’s Edge
  Foundation and Earth
  The End Of Eternity (not really in order apparently, kind of in its own time)
  Nemesis (1989) (not really in order, kind of on its own)

I got these lists from somewhere online years back, I think a stackoverflow site, but I'd avoid them because they also talk about spoilers even in listing the names of the books.

Many people (including me) think it’s best to start with the Foundation core set - Foundation, Foundation and Empire, and Second Foundation. Then you can move to the Robots Series - Caves of Steel, Naked Sun, and Robots of Dawn. Then circle back to the Foundation sequels, which ties them together a bit. It's important not to read Robots and Empire until you read the Foundation series.

And I'd top them off with The End Of Eternity, which although its in its own world (or is it?), it's ending ties indirectly to the whole series...