When I first discovered this science fiction series I decided to try and read the books not in chronological order (within the imaginary time-frame of the series), but rather in order of publication. This was on the basis that it makes sense to follow the unfolding story as the author conceived it, even if she starts in the middle and later goes back to fill in the preceding events.
However, I couldn't find anywhere on the internet which set out the information I wanted in a nice simple format. So, although I have not yet finished reading all the books, I am having a go myself at producing the kind of advice I would have found useful. Here is my suggestion for the order to tackle the Vorkosigan series.
I have tried to help out confused readers new to this series by mentioning some of the key people and planets, but hopefully without giving away too much of the plots.
(the following is not quite complete - a few more comments are to follow in due course as I read more books)
(Bujold's first four novels are set in the same imaginary universe, but the heroine/hero in each case is different. There are some obvious links between the stories, and some of the characters introduced here are developed much further in later books.)
Shards of Honour (1986)
Main character: Cordelia Naismith, from the sophisticated planet of Beta Colony. She meets (and eventually of course falls for) Aral Vorkosigan, has various wartime adventures, and discovers that his home planet of Barrayar, with its imperial government and military tradition, is not so bad after all.
The Warrior's Apprentice (1986)
Main character: Miles Vorkosigan, son of the above-mentioned Cordelia and Aral. This is a key book to read early on because it introduces so many characters and ideas, including Miles' cousin Ivan Vorpatril; the mercenary soldier Elli Quinn; Gregor the emperor of Barrayar; the 'Vor' hierarchical system; and the accidental establishment of the Dendarii mercenary fleet under the leadership of Miles (going by the assumed name of Admiral Naismith.)
Ethan of Athos (1986)
Main character: Ethan, from the backwater planet of Athos. He becomes unwittingly involved in an adventure of intrigue and mystery, helped out by Elli Quinn. Miles does not appear in person, but Elli makes frequent reference to a certain 'Admiral Naismith', her boss, whom she clearly idolises.
Falling Free (published 1988 - Nebula award 1988)
Main character: Leo Graf, engineer. Leo is sent to work on a space station and finds his sympathies are with the genetically modified workforce, called quaddies. This novel has no connection with the Vorkosigan family at all, but is set in an earlier century. (A quaddie turns up in the short story "Labyrinth" a few hundred years later.)
Brothers in Arms (1989)
Main character: Miles Vorkosigan, spending time on planet Earth having to juggle his real identity as a Vor Lord of Barrayar with his fake identity as a mercenary admiral. Various plots and counter-plots ensue, in the midst of which Miles encounters his clone brother, Mark.
Borders of Infinity (three short stories published together 1989)
A few pages here and there provide a setting in which Miles recalls three previous adventures: namely "The Mountains of Mourning", "Labyrinth" and "The Borders of Infinity"
The Vor Game (published 1990 - Hugo award 1991)
Main character: Miles Vorkosigan, starting out on his first official assignment. Soon he is haring around the galaxy, taking up leadership again of the Denedarii mercenaries and rescuing his friend Gregor (who also happens to be emperor of Barrayar) from life-threatening and politically awkward situations.
Barrayar (published 1991 - Hugo award 1992)
Main character: Cordelia, newly married to Aral Vorkosigan. An assassination attempt on his parents leaves the unborn Miles badly disabled. In the midst of rebellion and civil war on Barrayar, Cordelia struggles to protect the fragile life of her unborn son.
Mirror Dance (published 1994 - Hugo award 1995)
Main characters: Mark and Miles. The clone brother first encountered in Brothers in Arms attempts a mission of his own and finds himself caught up in family life and political intrigue on Barrayar, plus a rescue mission on the corrupt world of Jackson's Whole.
Main character: Miles Vorkosigan. At a relatively early stage in his career, Miles and Ivan are invited to the funeral of the empress of Cetaganda, Barrayar's traditional enemy. The Cetagandan Empire depends heavily on genetic manipulation to create an elite ruling body. Miles is unwittingly caught up in their internal power struggles.
Main character: Miles Vorkosigan. The events recorded in Mirror Dance have left their mark on Miles, who finds himself back home facing upheaval both in his career and his personal life. On top of that, he has to cope with the crisis faced by his boss and family friend, Simon Illyan, chief of Barrayaran Imperial Security.
Main characters: Ekaterin and Miles. The location is the planet Komarr, part of the Barrayaran Empire and an essential link between Barrayar and the rest of the galaxy. Ekaterin's husband works there. Miles is visiting to investigate a major industrial accident. Or is there some dastardly plot afoot? Of course there is.
A Civil Campaign (1999)
For a change, this story follows events from several people's point of view. Miles is still one of the key people, but his brother Mark is also heavily involved. Also, for a change, the plot does not revolve around life and death adventure, but around romance and courtship, business and politics on Miles' home planet of Barrayar.
Diplomatic Immunity (2002)
(If you read these in the order shown below, you get the story of Miles in order of internal chronology rather than publication. As an alternative to the suggested order above, this is a perfectly acceptable way to proceed.)
Cordelia's Honor - Shards of Honour & Barrayar
Young Miles - Warrior's Apprentice & "The Mountains of Mourning" (short story) & The Vor Game
Miles, Mystery and Mayhem - Cetaganda & Ethan of Athos & "Labyrinth" (short story)
Miles Errant - "The Borders of Infinity" (short story) & Brothers in Arms & Mirror Dance
(this leaves, in order of internal chronology: Memory, Komarr, A Civil Campaign & Diplomatic Immunity - which will no doubt be soon published as two further compilation volumes,)