Corey has suggested they are best read in publication order, (main novels in bold):
Leviathan Wakes (2011)
The Butcher of Anderson Station (2011)
Caliban's War (2012)
Gods of Risk (2012)
Drive (2012)
Abaddon's Gate (2013)
The Churn (2014)
Cibola Burn (2014)
Nemesis Games (2015)
The Vital Abyss (2015)
Babylon's Ashes (2016)
Untitled Expanse Novella (TBP)
Persepolis Rising (TBP)
However another good reading order, and IMO the better one, would be as follows (again main novels in bold, and note that I've jumbled short stories, novellas etc. under one title 'short story'):
Drive - Short story published November 27, 2012.
The Churn - Short story published April 29, 2014
The Butcher of Anderson Station - Short story published October 17, 2011
Leviathan Wakes - Novel published June 15, 2011
Caliban’s War - Novel published June 26, 2012
Gods of Risk - Short story published September 15, 2012
Abaddon’s Gate - Novel published June 4, 2013
The Vital Abyss - Short story published October 15, 2015
Cibola Burn - Novel published June 17, 2014
Nemesis Games - Novel published June 2, 2015
Babylon’s Ashes - Novel published December 6, 2016
Untitled Expanse Novella - Short story to be published
Persepolis Rising - Novel to be published 2017
Also note there are a few more being written, and several planned as well. And if you want to be pedantic, you could technically say that The Martian should be read first.
I personally think the second order is better as it makes a better read. It makes more sense and nothing happens which you need knowledge from another book that hasn't been read yet to understand. Although obviously I have no idea where the unpublished ones should go, so I've placed them chronologically.