First off, it's worth pointing out that David Weber is the Word of God on all of this, so take my opinion with a grain of salt.
The series started out as a novel series written by Weber beginning in 1992 with On Basilisk Station and continued with Weber as the sole creator up until roughly 2000 when Ashes of Victory was published. Two anthologies were also published - More than Honor and Worlds of Honor - in 1998 and 1999, respectively, with Weber writing stories and editing both anthologies, but none of the stories in those anthologies really add very much to the main story line.
However, between the publication of Ashes of Victory (or really, Echoes of Honor, the previous novel) and War of Honor, Weber kind of let the Honorverse get out of control. Just about everybody in the Baen bullpen took a crack at throwing new characters into the Honorverse and, while not all of them stuck, enough did that you have to pick up at least a couple of stories from the various anthologies to keep up with the overarching narrative.
The most important characters for the narrative, at least, right off the bat, are going to be the Zilwickis, Victor Cachat, and Abigail Hearns. They are featured in a couple of the short stories - "From the Highlands" in Changer of Worlds for the Zilwickis and Cachat, and "The Service of the Sword" in the anthology of the same name for Abigail Hearns.
After picking up those characters, you're ready for War of Honor. The Honorverse isn't done fragmenting, however. Two more spinoff series start pretty quickly after War of Honor: the Crown of Slaves books and the Saganami Island series. Luckily, since Weber (sometimes with Eric Flint) is writing all of them, you're pretty safe sticking with publication order - as long as you picked up those new supporting characters from the short stories.