Even though there are already three answers, one of which has been accepted, I'd like to give my preferred order. (Also, some books have been published after all these answers.)
It is almost internal chronological order, but with a few exceptions that I will justify.
First some remarks
1) Though Miles does not appear in "Ethan of Athos", this novel enters at a very precise place in the Saga and is best read exactly there
2) There is some risk of confusion about "Borders of Infinity" that needs clarification
It is at the title of a collection that contains four texts : three novellas called "Mountains of Mourning", "Labyrinth" and "The Borders of Infinity" (note the article "The") respectively, plus some "linking pages" often also called "Borders of Infinity". These linking pages have only be published in this collection, while all three novellas can be found in various omnibus editions.
The novel "Falling Free" is in my opinion one of the richest ones in the Saga, because of the deep humanity of some characters. It would be a great loss not to read it at all. But though chronologically it is the first one, I do not recommend to read it first. I think one should get acquainted with the world of Miles Vorkosigan first.
So skipping for the time being "Falling Free" proceed in chronological order starting from the second one. So:
Shards of Honor
then the novella "Mountains of Mourning"; if you have it only in the collection "Borders of Infinity", skip the "linking pages" that precede it and read it alone; if you have it in another edition, well, just read it
The Vor Game
Ethan of Athos (yes, precisely here)
At this point, before reading "Labyrinth, (or somewhat earlier,) would be a good time to read "Falling Free", but it is not yet necessary
If you do not have the collection "Borders of Infinity", read "Labyrinth" and then the novella "The Borders of Infinity", both of which appear in other places (omnibuses, for instance)
If you do have the collection "Borders of Infinity" then proceed to read it in order, first set of linking pages, skipping "Mountains of Mourning" that you have read long before (or rereading it, it is so beautiful !), proceeding to the next set of linking pages, then "Labyrinth", third set of linking pages, then the novella "The Borders of Infinity" and final set of linking pages.
Even though the events of the linking pages take place after those of "Brothers in Arms" they don't give any spoiler and are best read before it. This is the second exception to the chronological order. Then resume it
Brothers in Arms
A Civil Campaign
If you have not read "Falling Free" before, now it really is necessary to read it, otherwise you won't be able to fully enjoy your next read
Captain's Vorpatril Alliance
Cryoburn (which does not give any spoilers to Captain's Vorpatril Alliance - if you want you can read it first, as it was published first, it really does not matter)
Flowers of Vashnoi
Gentleman Jole and the Red Queen
If you can get hold of the collection "Dreamweaver Dilemma", know that the titular novella in the collection (not published anywhere else, to my knowledge) is also in the same Universe, and takes place even before "Falling Free". But it can be read at any time, the links with the remainder of the Saga are very weak. The only "plus" you get from it is an understanding of why Dubauer is such a common Betan name, as ruakh has already remarked in a comment to Is there any significance to the reuse of the name "Dubauer" in Shards of Honor and Diplomatic Immunity?
Also, if you cannot get hold of the collection "Borders of Infinity", just reading "Labyrinth" and the novella "The Borders of Infinity" will be quite OK, you won't miss much from the linking pages. There is just one little thing, there is one short discussion in the "linking pages" which is alluded to in a later book, and of course this allusion becomes incomprehensible. But most readers will just think they did read about it and forgot and won't worry about it. Only very few will try and check in the entire Saga to see where this comes from, and not find it !